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Abstract_algebra_0000 | Abstract_algebra | Fields and polynomials | Polynomials | 3 | [
"minimal polynomials"
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $5+2 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $5+2 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $5^{1/4}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}(c)$, where $c=\sqrt{15}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{15}$) | [
"x^2-10*x+29",
"x-(5+2*i)",
"x^4-5",
"x^4-16*x^2+4",
"x^2-2*c-8"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $-9+9 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $-9+9 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $2^{1/3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{13}+\sqrt{3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{13}+\sqrt{3}$ over $\mathbb{Q}(c)$, where $c=\sqrt{39}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{39}$) | [
"x^2+18*x+162",
"x-(9*i-9)",
"x^3-2",
"x^4-32*x^2+100",
"x^2-2*c-16"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] | Determine the minimal polynomial $f(x)$ of the following quantities:
(a) $-4+2 i$ over $\mathbb{R}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(b) $-4+2 i$ over $\mathbb{C}$, where $i=\sqrt{-1}$ $f(x)=$ [ANS]
(c) $3^{1/3}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(d) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}$ $f(x)=$ [ANS]
(e) $\sqrt{3}+\sqrt{5}$ over $\mathbb{Q}(c)$, where $c=\sqrt{15}$ $f(x)=$ [ANS]
(Your answer should be written using $c$, not $\sqrt{15}$) | [
"x^2+8*x+20",
"x-(2*i-4)",
"x^3-3",
"x^4-16*x^2+4",
"x^2-2*c-8"
] | [
"EX",
"EX",
"EX",
"EX",
"EX"
] | [
[],
[],
[],
[],
[]
] |
Abstract_algebra_0001 | Abstract_algebra | Fields and polynomials | Polynomials | 4 | [
"quotient fields",
"polynomial rings"
] | Let $t \in \mathbb{Q}[x]/(x^2-11)$ be a root of the irreducible polynomial $x^2-11 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(6-t)(7+2t)$: [ANS] $+$ [ANS] $t$
(c) $(7+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(6-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"121",
"20",
"5",
"93",
"28",
"0.24",
"0.04"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | Let $t \in \mathbb{Q}[x]/(x^2-3)$ be a root of the irreducible polynomial $x^2-3 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(2-t)(1+2t)$: [ANS] $+$ [ANS] $t$
(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(2-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"9",
"-4",
"3",
"13",
"4",
"2",
"1"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | Let $t \in \mathbb{Q}[x]/(x^2-5)$ be a root of the irreducible polynomial $x^2-5 \in \mathbb{Q}[x]$. Express each of the following elements in the form $u+wt$ with $u, w \in \mathbb{Q}$. The correct answers may involve fractions.
(a) $t^5$: [ANS] $+$ [ANS] $t$
(b) $(3-t)(1+2t)$: [ANS] $+$ [ANS] $t$
(c) $(1+2t)^2$: [ANS] $+$ [ANS] $t$
(d) $1/(3-t)$: [ANS] $+$ [ANS] $t$ | [
"0",
"25",
"-7",
"5",
"21",
"4",
"0.75",
"0.25"
] | [
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0002 | Abstract_algebra | Fields and polynomials | Polynomials | [
"polynomials"
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{11}$ such that f(0)=6, \quad f(7)=7, \quad f(8)=8 $f(x)=$ [ANS] | [
"x^3+2*x^2+6"
] | [
"EX"
] | [
[]
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{3}$ such that f(0)=2, \quad f(1)=1, \quad f(2)=2 $f(x)=$ [ANS] | [
"x^3+x^2+2"
] | [
"EX"
] | [
[]
] | Find a polynomial $f(x)$ of degree 3 over $\mathbb{Z}_{5}$ such that f(0)=3, \quad f(2)=2, \quad f(3)=3 $f(x)=$ [ANS] | [
"x^3+3*x^2+2*x+3"
] | [
"EX"
] | [
[]
] |
|
Abstract_algebra_0003 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"center of groups"
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"(e, x3)"
] | [
"UOL"
] | [
[]
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"e"
] | [
"EX"
] | [
[]
] | The center of a group $G$ is defined to be the set of all elements $x$ in $G$ such that $xy=yx$ for all $y$ in $G$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that lie in the center of $G$ and enter them as a comma-separated list. [ANS] | [
"(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0004 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"order of elements"
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x7)^2 & [ANS] \\ \hline (x8)^3 & [ANS] \\ \hline (x7)(x8) & [ANS] \\ \hline (x8)(x7) & [ANS] \\ \hline (x7)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x8$. Enter N if no such element exists. [ANS] | [
"x3",
"x11",
"x2",
"x4",
"x10",
"(e, x2, x4)",
"N",
"x11"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"EX"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x11)^2 & [ANS] \\ \hline (x2)^3 & [ANS] \\ \hline (x11)(x2) & [ANS] \\ \hline (x2)(x11) & [ANS] \\ \hline (x11)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x11$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x2$. Enter N if no such element exists. [ANS] | [
"x4",
"x2",
"x8",
"x10",
"x4",
"(e, x4, x5, x6, x7, x8, x9, x10, x11)",
"x4",
"x2"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"EX"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | The following is the group table of a group whose elements are $\lbrace e, x1, x2, \ldots, x11 \rbrace$, where $e$ is the identity:
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
(a) Express each of the following elements in terms of one element from $\lbrace e, x1, x2, \ldots, x11 \rbrace$.
$\begin{array}{cc}\hline (x7)^2 & [ANS] \\ \hline (x3)^3 & [ANS] \\ \hline (x7)(x3) & [ANS] \\ \hline (x3)(x7) & [ANS] \\ \hline (x7)^{-1} & [ANS] \\ \hline \end{array}$
(b) Find all elements $x$ such that $x^3=e$. Enter N if no such element exists. [ANS]
(c) Find all elements $x$ such that $x^2=x7$. Enter N if no such element exists. [ANS]
(d) Find all elements $x$ such that $x^3=x3$. Enter N if no such element exists. [ANS] | [
"x2",
"x3",
"x10",
"x10",
"x11",
"(e, x2, x4)",
"N",
"(x1, x3, x5)"
] | [
"EX",
"EX",
"EX",
"EX",
"EX",
"UOL",
"EX",
"UOL"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0005 | Abstract_algebra | Groups | Group axioms | 3 | [
"group tables",
"commutativity"
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x11 & x10 & x9 & x8 & x7 & x3 & x2 & x1 & e & x5 & x4 \\ \hline x7 & x7 & x6 & x11 & x10 & x9 & x8 & x4 & x3 & x2 & x1 & e & x5 \\ \hline x8 & x8 & x7 & x6 & x11 & x10 & x9 & x5 & x4 & x3 & x2 & x1 & e \\ \hline x9 & x9 & x8 & x7 & x6 & x11 & x10 & e & x5 & x4 & x3 & x2 & x1 \\ \hline x10 & x10 & x9 & x8 & x7 & x6 & x11 & x1 & e & x5 & x4 & x3 & x2 \\ \hline x11 & x11 & x10 & x9 & x8 & x7 & x6 & x2 & x1 & e & x5 & x4 & x3 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS] | [
"(e, x3, x7, x10)"
] | [
"UOL"
] | [
[]
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & e & x3 & x2 & x6 & x7 & x4 & x5 & x11 & x10 & x9 & x8 \\ \hline x2 & x2 & x3 & e & x1 & x7 & x6 & x5 & x4 & x9 & x8 & x11 & x10 \\ \hline x3 & x3 & x2 & x1 & e & x5 & x4 & x7 & x6 & x10 & x11 & x8 & x9 \\ \hline x4 & x4 & x7 & x5 & x6 & x11 & x8 & x10 & x9 & x2 & x1 & x3 & e \\ \hline x5 & x5 & x6 & x4 & x7 & x9 & x10 & x8 & x11 & x1 & x2 & e & x3 \\ \hline x6 & x6 & x5 & x7 & x4 & x8 & x11 & x9 & x10 & x3 & e & x2 & x1 \\ \hline x7 & x7 & x4 & x6 & x5 & x10 & x9 & x11 & x8 & e & x3 & x1 & x2 \\ \hline x8 & x8 & x10 & x11 & x9 & x1 & x3 & x2 & e & x7 & x5 & x4 & x6 \\ \hline x9 & x9 & x11 & x10 & x8 & x3 & x1 & e & x2 & x4 & x6 & x7 & x5 \\ \hline x10 & x10 & x8 & x9 & x11 & x2 & e & x1 & x3 & x6 & x4 & x5 & x7 \\ \hline x11 & x11 & x9 & x8 & x10 & e & x2 & x3 & x1 & x5 & x7 & x6 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x11$ and enter them as a comma-separated list. [ANS] | [
"(e, x4, x11)"
] | [
"UOL"
] | [
[]
] | Two elements $x$, $y$ of a group are said to commute with each other if $xy=yx$.
Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline e & e & x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 & x10 & x11 \\ \hline x1 & x1 & x2 & x3 & x4 & x5 & e & x7 & x8 & x9 & x10 & x11 & x6 \\ \hline x2 & x2 & x3 & x4 & x5 & e & x1 & x8 & x9 & x10 & x11 & x6 & x7 \\ \hline x3 & x3 & x4 & x5 & e & x1 & x2 & x9 & x10 & x11 & x6 & x7 & x8 \\ \hline x4 & x4 & x5 & e & x1 & x2 & x3 & x10 & x11 & x6 & x7 & x8 & x9 \\ \hline x5 & x5 & e & x1 & x2 & x3 & x4 & x11 & x6 & x7 & x8 & x9 & x10 \\ \hline x6 & x6 & x7 & x8 & x9 & x10 & x11 & e & x1 & x2 & x3 & x4 & x5 \\ \hline x7 & x7 & x8 & x9 & x10 & x11 & x6 & x1 & x2 & x3 & x4 & x5 & e \\ \hline x8 & x8 & x9 & x10 & x11 & x6 & x7 & x2 & x3 & x4 & x5 & e & x1 \\ \hline x9 & x9 & x10 & x11 & x6 & x7 & x8 & x3 & x4 & x5 & e & x1 & x2 \\ \hline x10 & x10 & x11 & x6 & x7 & x8 & x9 & x4 & x5 & e & x1 & x2 & x3 \\ \hline x11 & x11 & x6 & x7 & x8 & x9 & x10 & x5 & e & x1 & x2 & x3 & x4 \\ \hline \end{array}$
Using this group table, determine the elements that commute with $x7$ and enter them as a comma-separated list. [ANS] | [
"(e, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0006 | Abstract_algebra | Groups | Subgroups | 3 | [
"subgroups"
] | Find all elements of the subgroup $\langle 12 \rangle$ in $\mathbb{Z}_{84}$. [ANS]
Find all elements of the subgroup $\langle 9 \rangle$ in $\mathbb{Z}_{72}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 12, 24, 36, 48, 60, 72)",
"(0, 9, 18, 27, 36, 45, 54, 63)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements of the subgroup $\langle 6 \rangle$ in $\mathbb{Z}_{42}$. [ANS]
Find all elements of the subgroup $\langle 12 \rangle$ in $\mathbb{Z}_{96}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 6, 12, 18, 24, 30, 36)",
"(0, 12, 24, 36, 48, 60, 72, 84)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements of the subgroup $\langle 8 \rangle$ in $\mathbb{Z}_{56}$. [ANS]
Find all elements of the subgroup $\langle 11 \rangle$ in $\mathbb{Z}_{77}$. [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(0, 8, 16, 24, 32, 40, 48)",
"(0, 11, 22, 33, 44, 55, 66)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0007 | Abstract_algebra | Groups | Subgroups | 3 | [
"group tables",
"order of elements"
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\ \hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\ \hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\ \hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\ \hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\ \hline x_{6} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} \\ \hline x_{7} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{8} & x_{4} & x_{3} & x_{2} & x_{1} & e & x_{5} \\ \hline x_{8} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & x_{9} & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} & e \\ \hline x_{9} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{10} & e & x_{5} & x_{4} & x_{3} & x_{2} & x_{1} \\ \hline x_{10} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{11} & x_{1} & e & x_{5} & x_{4} & x_{3} & x_{2} \\ \hline x_{11} & x_{11} & x_{10} & x_{9} & x_{8} & x_{7} & x_{6} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{3} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{7} & [ANS] \\ \hline x_{8} & [ANS] \\ \hline x_{7} x_{8} & [ANS] \\ \hline (x_{7})^{2}(x_{8}) & [ANS] \\ \hline (x_{7})^{-1}(x_{8}) & [ANS] \\ \hline \end{array}$ | [
"4",
"4",
"3",
"4",
"6"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & e & x_{3} & x_{2} & x_{6} & x_{7} & x_{4} & x_{5} & x_{11} & x_{10} & x_{9} & x_{8} \\ \hline x_{2} & x_{2} & x_{3} & e & x_{1} & x_{7} & x_{6} & x_{5} & x_{4} & x_{9} & x_{8} & x_{11} & x_{10} \\ \hline x_{3} & x_{3} & x_{2} & x_{1} & e & x_{5} & x_{4} & x_{7} & x_{6} & x_{10} & x_{11} & x_{8} & x_{9} \\ \hline x_{4} & x_{4} & x_{7} & x_{5} & x_{6} & x_{11} & x_{8} & x_{10} & x_{9} & x_{2} & x_{1} & x_{3} & e \\ \hline x_{5} & x_{5} & x_{6} & x_{4} & x_{7} & x_{9} & x_{10} & x_{8} & x_{11} & x_{1} & x_{2} & e & x_{3} \\ \hline x_{6} & x_{6} & x_{5} & x_{7} & x_{4} & x_{8} & x_{11} & x_{9} & x_{10} & x_{3} & e & x_{2} & x_{1} \\ \hline x_{7} & x_{7} & x_{4} & x_{6} & x_{5} & x_{10} & x_{9} & x_{11} & x_{8} & e & x_{3} & x_{1} & x_{2} \\ \hline x_{8} & x_{8} & x_{10} & x_{11} & x_{9} & x_{1} & x_{3} & x_{2} & e & x_{7} & x_{5} & x_{4} & x_{6} \\ \hline x_{9} & x_{9} & x_{11} & x_{10} & x_{8} & x_{3} & x_{1} & e & x_{2} & x_{4} & x_{6} & x_{7} & x_{5} \\ \hline x_{10} & x_{10} & x_{8} & x_{9} & x_{11} & x_{2} & e & x_{1} & x_{3} & x_{6} & x_{4} & x_{5} & x_{7} \\ \hline x_{11} & x_{11} & x_{9} & x_{8} & x_{10} & e & x_{2} & x_{3} & x_{1} & x_{5} & x_{7} & x_{6} & x_{4} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{11} & [ANS] \\ \hline x_{2} & [ANS] \\ \hline x_{11} x_{2} & [ANS] \\ \hline (x_{11})^{2}(x_{2}) & [ANS] \\ \hline (x_{11})^{-1}(x_{2}) & [ANS] \\ \hline \end{array}$ | [
"3",
"2",
"3",
"3",
"3"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] | Consider the group whose group table is given as follows ($e$ is the identity element):
$\begin{array}{ccccccccccccc}\hline & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline e & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} \\ \hline x_{1} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} \\ \hline x_{2} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} \\ \hline x_{3} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} \\ \hline x_{4} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} \\ \hline x_{5} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} \\ \hline x_{6} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & e & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\ \hline x_{7} & x_{7} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & e \\ \hline x_{8} & x_{8} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{2} & x_{3} & x_{4} & x_{5} & e & x_{1} \\ \hline x_{9} & x_{9} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{3} & x_{4} & x_{5} & e & x_{1} & x_{2} \\ \hline x_{10} & x_{10} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{4} & x_{5} & e & x_{1} & x_{2} & x_{3} \\ \hline x_{11} & x_{11} & x_{6} & x_{7} & x_{8} & x_{9} & x_{10} & x_{5} & e & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline \end{array}$
Determine the order of the following elements and complete the table:
$\begin{array}{cc}\hline x & order(x) \\ \hline x_{7} & [ANS] \\ \hline x_{3} & [ANS] \\ \hline x_{7} x_{3} & [ANS] \\ \hline (x_{7})^{2}(x_{3}) & [ANS] \\ \hline (x_{7})^{-1}(x_{3}) & [ANS] \\ \hline \end{array}$ | [
"6",
"2",
"6",
"6",
"6"
] | [
"NV",
"NV",
"NV",
"NV",
"NV"
] | [
[],
[],
[],
[],
[]
] |
Abstract_algebra_0008 | Abstract_algebra | Groups | Subgroups | 3 | [
"subgroups",
"generators"
] | Find all elements $x$ in $U(198)$ such that $\langle x \rangle$=$\langle 17 \rangle$: [ANS]
Also find all elements $x$ in $U(189)$ such that $\langle x \rangle$=$\langle 4 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(17, 161, 107, 35)",
"(4, 16, 67, 79, 130, 142)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements $x$ in $U(164)$ such that $\langle x \rangle$=$\langle 3 \rangle$: [ANS]
Also find all elements $x$ in $U(205)$ such that $\langle x \rangle$=$\langle 3 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(3, 27, 79, 55)",
"(3, 27, 38, 137)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find all elements $x$ in $U(175)$ such that $\langle x \rangle$=$\langle 6 \rangle$: [ANS]
Also find all elements $x$ in $U(190)$ such that $\langle x \rangle$=$\langle 61 \rangle$: [ANS]
For both parts, enter your answers as comma-separated lists. | [
"(6, 41, 111, 146)",
"(61, 111, 161, 131, 101, 81)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0009 | Abstract_algebra | Groups | Subgroups | 4 | [
"subgroups",
"generators"
] | Find one pair of elements $x,$ $y$ in $U(14)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(14)$ and that $\langle x,y \rangle=U(14)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"9",
"13"
] | [
"NV",
"NV"
] | [
[]
] | Find one pair of elements $x,$ $y$ in $U(7)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(7)$ and that $\langle x,y \rangle=U(7)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"(2, 6)"
] | [
"OL"
] | [
[]
] | Find one pair of elements $x,$ $y$ in $U(8)$ such that $\langle x \rangle$ and $\langle y \rangle$ are proper subgroups of $U(8)$ and that $\langle x,y \rangle=U(8)$. Be sure that $x<y$ with $1 \leq x, y<n$.
$(x,y)=($ [ANS] $,$ [ANS] $)$ | [
"(3, 5)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0010 | Abstract_algebra | Groups | Cyclic groups | 3 | [
"cyclic groups",
"generators"
] | Determine all generators of $\mathbb{Z}_{21}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20)"
] | [
"UOL"
] | [
[]
] | Determine all generators of $\mathbb{Z}_{15}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 4, 7, 8, 11, 13, 14)"
] | [
"UOL"
] | [
[]
] | Determine all generators of $\mathbb{Z}_{18}$. Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 7, 11, 13, 17)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0011 | Abstract_algebra | Groups | Cyclic groups | 3 | [
"cyclic groups",
"order of elements"
] | (a) Find all elements in $\{ cyclic(143) \}$ of order $13$. [ANS]
(b) Find all elements in the subgroup $\langle 13 \rangle$ of $\{ cyclic(143) \}$. [ANS] | [
"(11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132)",
"(0, 13, 26, 39, 52, 65, 78, 91, 104, 117, 130)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all elements in $\{ cyclic(51) \}$ of order $3$. [ANS]
(b) Find all elements in the subgroup $\langle 3 \rangle$ of $\{ cyclic(51) \}$. [ANS] | [
"(17, 34)",
"(0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all elements in $\{ cyclic(55) \}$ of order $5$. [ANS]
(b) Find all elements in the subgroup $\langle 5 \rangle$ of $\{ cyclic(55) \}$. [ANS] | [
"(11, 22, 33, 44)",
"(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0012 | Abstract_algebra | Groups | Cyclic groups | 4 | [
"cyclic groups",
"generators",
"subgroups"
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{35}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{35}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 5, 7, 10, 14, 15, 20, 21, 25, 28, 30)"
] | [
"UOL"
] | [
[]
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{33}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{33}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 3, 6, 9, 11, 12, 15, 18, 21, 22, 24, 27, 30)"
] | [
"UOL"
] | [
[]
] | Find all elements $x_1, x_2, x_3,...$ in $\mathbb{Z}_{28}$ such that each $\langle x_i \rangle$ is a proper subgroup of $\mathbb{Z}_{28}$. Enter your answer as a comma-separated list. [ANS] | [
"(0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0013 | Abstract_algebra | Groups | Cyclic groups | 6 | [
"cyclic groups",
"generators",
"subgroups"
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{1309}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{1309}$, and that $\langle x, y \rangle=\mathbb{Z}_{1309}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 1309 is a product of three primes. | [
"(17, 11)"
] | [
"UOL"
] | [
[]
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{114}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{114}$, and that $\langle x, y \rangle=\mathbb{Z}_{114}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 114 is a product of three primes. | [
"(2, 19)"
] | [
"UOL"
] | [
[]
] | Find one pair $(x, y)$ of elements of $\mathbb{Z}_{195}$ such that both $\langle x \rangle, \langle y \rangle$ are proper subgroups of $\mathbb{Z}_{195}$, and that $\langle x, y \rangle=\mathbb{Z}_{195}$.
$x=$ [ANS]
$y=$ [ANS]
HINT: 195 is a product of three primes. | [
"(5, 13)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0014 | Abstract_algebra | Groups | Cyclic groups | 4 | [
"cyclic groups",
"order of groups",
"order of elements",
"subgroups"
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=18$ and $\textrm{ord}(y)=24$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 2, 3, 6)"
] | [
"OL"
] | [
[]
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=12$ and $\textrm{ord}(y)=15$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 3)"
] | [
"OL"
] | [
[]
] | Let $x,$ $y$ be elements of a group G. If $\textrm{ord}(x)=14$ and $\textrm{ord}(y)=21$, what are the possible values for the order of $\langle x \rangle \cap \langle y \rangle$? Enter your answer as a list of numbers separated by commas, or a single number if there is only one possible value. [ANS] | [
"(1, 7)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0015 | Abstract_algebra | Groups | Cyclic groups | 2 | [
"cyclic groups",
"order of elements"
] | Determine the order of every element of $\mathbb{Z}_{21}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 21, 21, 7, 21, 21, 7, 3, 21, 7, 21, 21, 7, 21, 3, 7, 21, 21, 7, 21, 21)"
] | [
"OL"
] | [
[]
] | Determine the order of every element of $\mathbb{Z}_{15}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 15, 15, 5, 15, 3, 5, 15, 15, 5, 3, 15, 5, 15, 15)"
] | [
"OL"
] | [
[]
] | Determine the order of every element of $\mathbb{Z}_{18}$. Enter your answer as a comma-separated ORDERED list of this form:
$\textrm{ord}(0),$ $\textrm{ord}(1),$... $\textrm{ord}(j),$... [ANS] | [
"(1, 18, 9, 6, 9, 18, 3, 18, 9, 2, 9, 18, 3, 18, 9, 6, 9, 18)"
] | [
"OL"
] | [
[]
] |
Abstract_algebra_0017 | Abstract_algebra | Groups | Product of groups | 4 | [
"product of groups",
"generators"
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{176}$ such that $\mathbb{Z}_{176}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(16, 11)"
] | [
"UOL"
] | [
[]
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{75}$ such that $\mathbb{Z}_{75}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(3, 25)"
] | [
"UOL"
] | [
[]
] | Find a pair of NON-IDENTITY elements $A, B$ in $\mathbb{Z}_{77}$ such that $\mathbb{Z}_{77}$ is isomorphic to $\langle A \rangle \times \langle B \rangle$.
$A, B=$ [ANS] | [
"(7, 11)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0018 | Abstract_algebra | Groups | Product of groups | 4 | [
"products of groups",
"generators"
] | Find a pair of elements $A, B$ in $U(63)$ such that $U(63)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 63$. $A, B=$ [ANS] | [
"(10, 29)"
] | [
"UOL"
] | [
[]
] | Find a pair of elements $A, B$ in $U(69)$ such that $U(69)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 69$. $A, B=$ [ANS] | [
"(47, 28)"
] | [
"UOL"
] | [
[]
] | Find a pair of elements $A, B$ in $U(91)$ such that $U(91)$ is isomorphic to $\langle A \rangle \times \langle B \rangle$. Be sure to enter your answer as a comma-separated list of two POSITIVE integers $< 91$. $A, B=$ [ANS] | [
"(66, 15)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0019 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets"
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{80}$, $H=10 \mathbb{Z}_{80}$, $g=53 \pmod{80}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<k\right>$ $g$ $=k$ [ANS] | [
"(53, 63, 73, 3, 13, 23, 33, 43)",
"(1, -1, k, -k)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{56}$, $H=14 \mathbb{Z}_{56}$, $g=12 \pmod{56}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<j\right>$ $g$ $=-i$ [ANS] | [
"(12, 26, 40, 54)",
"(-i, i, k, -k)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Let $H$ be a subgroup of $G$. For any element $g$ in $G$. recall that the left coset $gH$ is by definition the subset:
$gH:=\lbrace gh: h \in H \rbrace$
For each of the following triples of $(G, H, g)$, write down the elements of the left coset $gH$ (enter your answer as a comma-separated list):
(i) $G=\mathbb{Z}_{55}$, $H=11 \mathbb{Z}_{55}$, $g=19 \pmod{55}$ [ANS]
(ii) $G=$ the quaternion group $Q_8=\lbrace 1,-1, i,-i, j,-j, k,-k \rbrace$ $H=\left<j\right>$ $g$ $=k$ [ANS] | [
"(19, 30, 41, 52, 8)",
"(k, -k, i, -i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0020 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets",
"coset representatives"
] | Find a complete set of coset representatives of the subgroup $\langle 19 \rangle$ in $U(56)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 56$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 8 \rangle$ in $U(51)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 51$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 5, 11, 29)",
"(1, 5, 11, 19)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find a complete set of coset representatives of the subgroup $\langle 4 \rangle$ in $U(35)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 35$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 20 \rangle$ in $U(63)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 63$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 2, 3, 6)",
"(1, 2, 4, 5, 8, 10)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | Find a complete set of coset representatives of the subgroup $\langle 37 \rangle$ in $U(42)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 42$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS]
Find a complete set of coset representatives of the subgroup $\langle 21 \rangle$ in $U(52)$. Enter your answer as a comma separated list; make sure that EACH coset representative you enter
$\ast$ is $>0$ and $< 52$, and $\ast$ is the smallest possible value in this range, i.e. if you enter the value $A$, there is not another value $0<B<$ with $B<A$ such that $A, B$ represents the same coset. [ANS] | [
"(1, 5, 11, 13)",
"(1, 3, 7, 9, 27, 29)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0022 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 6 | [
"Lagrange theorem"
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=35$ and $\#G=350$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(70, 175)"
] | [
"UOL"
] | [
[]
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=22$ and $\#G=308$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(44, 154)"
] | [
"UOL"
] | [
[]
] | Let $H$ be a proper subgroup of a group $G$, and let $K$ be a proper subgroup of $H$. If $\#K=26$ and $\#G=260$, what are the possible orders of $H$? Enter your answer as a comma separated list. [ANS] | [
"(52, 130)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0023 | Abstract_algebra | Groups | Cosets, Lagrange's theorem, and normality | 2 | [
"cosets"
] | (a) Determine all elements of the coset $4+\langle 10 \rangle$ in the group $\mathbb{Z}_{14}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 14$. [ANS]
(b) Determine all elements of the coset $1 \langle 26 \rangle$ in the group $U(57)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 57$. [ANS] | [
"(4, 6, 8, 10, 12, 0, 2)",
"(26, 49, 20, 7, 11, 1)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Determine all elements of the coset $1+\langle 6 \rangle$ in the group $\mathbb{Z}_{6}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 6$. [ANS]
(b) Determine all elements of the coset $2 \langle 26 \rangle$ in the group $U(57)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 57$. [ANS] | [
"1",
"(52, 41, 40, 14, 22, 2)"
] | [
"NV",
"UOL"
] | [
[],
[]
] | (a) Determine all elements of the coset $2+\langle 6 \rangle$ in the group $\mathbb{Z}_{9}$. Enter your answer as a comma separated list; make sure that each element you enter is $\geq 0$ and $< 9$. [ANS]
(b) Determine all elements of the coset $1 \langle 41 \rangle$ in the group $U(72)$. Enter your answer as a comma separated list; make sure that each element you enter is $>0$ and $< 72$. [ANS] | [
"(2, 5, 8)",
"(41, 25, 17, 49, 65, 1)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0024 | Abstract_algebra | Groups | Homomorphisms | 6 | [
"group homomorphisms",
"cyclic groups",
"order of groups",
"image"
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{245} \rightarrow \mathbb{Z}_{175}$
such that the image of $f$ has size exactly $7$.
First, since $245 \equiv 0$ $\pmod{245}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(245) \\ & \equiv & 245 f(1) \pmod{175} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{175}$ so $0 \equiv 175 f(1) \pmod{175}$.
Thus $\gcd(245,175) f(1) \equiv 0$ $\pmod{175}$ i.e. $35 f(1) \equiv 0$ $\pmod{175}$ whence $(\ast)$ $f(1)=5 u$ for some $u$ in $\mathbb{Z}_{175}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{175}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $7$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $7$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 25, 50, 75, 100, 125, 150)",
"(25, 50, 75, 100, 125, 150)",
"6"
] | [
"OL",
"OL",
"NV"
] | [
[],
[],
[]
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{28} \rightarrow \mathbb{Z}_{98}$
such that the image of $f$ has size exactly $2$.
First, since $28 \equiv 0$ $\pmod{28}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(28) \\ & \equiv & 28 f(1) \pmod{98} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{98}$ so $0 \equiv 98 f(1) \pmod{98}$.
Thus $\gcd(28,98) f(1) \equiv 0$ $\pmod{98}$ i.e. $14 f(1) \equiv 0$ $\pmod{98}$ whence $(\ast)$ $f(1)=7 u$ for some $u$ in $\mathbb{Z}_{98}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{98}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $2$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $2$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 49)",
"49",
"1"
] | [
"OL",
"NV",
"NV"
] | [
[],
[],
[]
] | In this problem we determine the number of group homomorphisms
$f: \mathbb{Z}_{45} \rightarrow \mathbb{Z}_{75}$
such that the image of $f$ has size exactly $3$.
First, since $45 \equiv 0$ $\pmod{45}$, we have
$\begin{array}{llllllll} 0 & \equiv & f(0) && \text{property of homomorphisms} \\ & \equiv & f(45) \\ & \equiv & 45 f(1) \pmod{75} && \text{property of homomorphisms.} \end{array}$ On the other hand, $f(1)$ is an element of $\mathbb{Z}_{75}$ so $0 \equiv 75 f(1) \pmod{75}$.
Thus $\gcd(45,75) f(1) \equiv 0$ $\pmod{75}$ i.e. $15 f(1) \equiv 0$ $\pmod{75}$ whence $(\ast)$ $f(1)=5 u$ for some $u$ in $\mathbb{Z}_{75}$.
On the other hand, the image of $f$ is a subgroup of the cyclic group $\mathbb{Z}_{75}$. But a cyclic group has at most ONE subgroup of any given order, so if the image of $f$ has size $3$ then the elements of the image of $f$ must be [ANS] (please enter your answer as an ORDERED list)
Combine this with $(\ast)$ and we see that in order for the image of $f$ to have size $3$, the choices for $f(1)$ are [ANS] (please enter your answer as an ORDERED list)
Consequently, the number of such functions $f$ is [ANS]. Please enter your answer as a number. | [
"(0, 25, 50)",
"(25, 50)",
"2"
] | [
"OL",
"OL",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0025 | Abstract_algebra | Groups | Group actions | 6 | [
"group actions",
"orbit-stabilizer theorem"
] | Let $G$ be a finite group of order $77$ acting on a finite set $S$ of size $11$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 11)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $7$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 3, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $15$ acting on a finite set $S$ of size $5$. What are the possible values for the NUMBER of orbits of this $G$-action? Enter your answer as a comma-separated list. [ANS] | [
"(1, 3, 5)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0026 | Abstract_algebra | Groups | Group actions | 6 | [
"group actions",
"orbit-stabilizer theorem"
] | Let $G$ be a finite group of order $35$ acting on a finite set $S$ of size $34$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 5, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $49$ acting on a finite set $S$ of size $17$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 7)"
] | [
"UOL"
] | [
[]
] | Let $G$ be a finite group of order $36$ acting on a finite set $S$ of size $23$. What are the possible values for the size of the orbit of an element of $S$? Enter your answer as a comma-separated list. [ANS] | [
"(1, 2, 3, 4, 6, 9, 12, 18)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0027 | Abstract_algebra | Groups | Group actions | 4 | [
"group actions",
"conjugation",
"conjugacy classes"
] | In this problem we determine the conjugacy class of the elements $a^{6} b^0$ and $a^{6} b^1$ in the dihedral group $D_{11}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{6} b^0? & is a^i b^1conjugate to a^{6} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 7 & [ANS] & [ANS] \\ \hline 8 & [ANS] & [ANS] \\ \hline 9 & [ANS] & [ANS] \\ \hline 10 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"Y",
"Y",
"Y",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y",
"N",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{6}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"N",
"Y",
"Y",
"N",
"N",
"N",
"Y",
"N",
"N",
"Y",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | In this problem we determine the conjugacy class of the elements $a^{5} b^0$ and $a^{5} b^1$ in the dihedral group $D_{8}$.
Complete the table by entering Y or N in each entry:
$\begin{array}{ccc}\hline i & is a^i b^0conjugate to a^{5} b^0? & is a^i b^1conjugate to a^{5} b^1? \\ \hline 0 & [ANS] & [ANS] \\ \hline 1 & [ANS] & [ANS] \\ \hline 2 & [ANS] & [ANS] \\ \hline 3 & [ANS] & [ANS] \\ \hline 4 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 7 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"N",
"N",
"Y",
"N",
"N",
"Y",
"Y",
"N",
"N",
"Y",
"Y",
"N",
"N",
"N",
"Y"
] | [
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0028 | Abstract_algebra | Rings | Ring axioms | 2 | [
"ring axioms"
] | Let $a=148$ and $b=157$ be elements in the ring $\mathbb{Z}_{175}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 175$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"130",
"166",
"136"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Let $a=40$ and $b=62$ be elements in the ring $\mathbb{Z}_{63}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 63$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"39",
"41",
"23"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Let $a=56$ and $b=60$ be elements in the ring $\mathbb{Z}_{75}$. Evaluate the following expressions. For each one, enter your answer as an integer $0 \leq n < 75$.
(a) $a+b=$ [ANS]
(b) $a-b=$ [ANS]
(c) $a \times b=$ [ANS] | [
"41",
"71",
"60"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0029 | Abstract_algebra | Rings | Ring axioms | 3 | [
"ring axioms"
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6,7,8,9} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {1,2,4,5,8,9} \rbrace
B=\lbrace {1,2,7,9} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"512",
"(1, 2, 4, 5, 8, 9)",
"(4, 5, 7, 8)",
"(1, 2, 9)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {2,3,4,5,6} \rbrace
B=\lbrace {3,4,5} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"64",
"(2, 3, 4, 5, 6)",
"(2, 6)",
"(3, 4, 5)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] | For any set $X$, denote by $P^X$ the set of all subsets of $X$ (including the empty set $\emptyset$ and $X$ itself). This is called the power set of $X$.
If $A, B$ are elements of $P^X$, define
$A+B:=(A-B) \cup (B-A)$
$A \times B:=A \cap B$
FACT: $P^X$ together with these two operations forms a commutative ring with a multiplicative identity.
For the rest of this exercise, take $X$ to be the set $\lbrace {1,2,3,4,5,6,7} \rbrace$.
(a) How many elements are there in $P^X$? [ANS]
Now let $A,B$ be two subsets of $X$ defined as follows:
A=\lbrace {1,3,4,6,7} \rbrace
B=\lbrace {1,2,4} \rbrace Enter the elements of each set below as a comma-separated list. Input N for the empty set.
(b) What is the additive inverse of the subset $A$? $\lbrace$ [ANS] $\rbrace$
(c) What is $A+B$? $\lbrace$ [ANS] $\rbrace$
(d) What is $A \times B$? $\lbrace$ [ANS] $\rbrace$ | [
"128",
"(1, 3, 4, 6, 7)",
"(2, 3, 6, 7)",
"(1, 4)"
] | [
"NV",
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[],
[]
] |
Abstract_algebra_0030 | Abstract_algebra | Rings | Ring axioms | 3 | [
"ring axioms",
"inverse"
] | (a) Find the multiplicative inverse of $38$ in $\mathbb{Z}_{39}$. [ANS]
(b) Find the multiplicative inverse of $30$ in $\mathbb{Z}_{31}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"38",
"30",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] | (a) Find the multiplicative inverse of $8$ in $\mathbb{Z}_{9}$. [ANS]
(b) Find the multiplicative inverse of $46$ in $\mathbb{Z}_{47}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"8",
"46",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] | (a) Find the multiplicative inverse of $19$ in $\mathbb{Z}_{20}$. [ANS]
(b) Find the multiplicative inverse of $32$ in $\mathbb{Z}_{33}$. [ANS]
(c) In general, what is the multiplicative inverse of $(n-1)$ in $\mathbb{Z}_{n}$? [ANS] | [
"19",
"32",
"n-1"
] | [
"NV",
"NV",
"EX"
] | [
[],
[],
[]
] |
Abstract_algebra_0031 | Abstract_algebra | Rings | Units and zero divisors | 4 | [
"commutativity",
"zero-divisors"
] | Denote by $R$ the set of all functions from the set $\lbrace-8, 8, 9, 10 \rbrace$ to the ring $\mathbb{Z}_{45}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-8 & [ANS] \\ \hline 8 & [ANS] \\ \hline 9 & [ANS] \\ \hline 10 & [ANS] \\ \hline \end{array}$ | [
"Y",
"331776",
"(3, 0, 0, 0)"
] | [
"TF",
"NV",
"UOL"
] | [
[],
[],
[]
] | Denote by $R$ the set of all functions from the set $\lbrace-10, 2, 8, 10 \rbrace$ to the ring $\mathbb{Z}_{30}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-10 & [ANS] \\ \hline 2 & [ANS] \\ \hline 8 & [ANS] \\ \hline 10 & [ANS] \\ \hline \end{array}$ | [
"Y",
"4096",
"(2, 0, 0, 0)"
] | [
"TF",
"NV",
"UOL"
] | [
[],
[],
[]
] | Denote by $R$ the set of all functions from the set $\lbrace-8,-1, 2, 8 \rbrace$ to the ring $\mathbb{Z}_{18}$. FACT: $R$ becomes a ring under the following operations:
$f+g: a \mapsto f(a)+g(a), f*g: a \mapsto f(a)g(a)$ (a) Is $R$ a commutative ring? (Y/N) [ANS]
(b) How many units are there in $R$? [ANS]
(c) Give an example of a non-zero $f \in R$ that is a zero-divisor by filling in the following table:
$\begin{array}{cc}\hline x & f(x) \\ \hline-8 & [ANS] \\ \hline-1 & [ANS] \\ \hline 2 & [ANS] \\ \hline 8 & [ANS] \\ \hline \end{array}$ | [
"TF",
"1296",
"(2, 0, 0, 0)"
] | [
"EX",
"NV",
"UOL"
] | [
[],
[],
[]
] |
Abstract_algebra_0032 | Abstract_algebra | Rings | Units and zero divisors | 3 | [
"characteristic"
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{46} \times \mathbb{Z}_{36}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{38} \times \mathbb{Z}_{21}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{35}$. [ANS] | [
"828",
"798",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{6} \times \mathbb{Z}_{56}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{10} \times \mathbb{Z}_{21}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{57}$. [ANS] | [
"168",
"210",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | (a) Determine the characteristic of the ring $\mathbb{Z}_{20} \times \mathbb{Z}_{18}$. [ANS]
(b) Determine the characteristic of the ring $\mathbb{Z}_{34} \times \mathbb{Z}_{49}$. [ANS]
(c) Determine the characteristic of the ring $\mathbb{Z} \times \mathbb{Z}_{55}$. [ANS] | [
"180",
"1666",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0033 | Abstract_algebra | Rings | Units and zero divisors | 4 | [
"characteristic",
"integral domains"
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{36} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{44} \times \mathbb{Z}_{19} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{41} \times \mathbb{Z}_{41} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{38}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"836",
"N",
"41",
"N",
"0",
"Y",
"38",
"N"
] | [
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{56} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{21} \times \mathbb{Z}_{57} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{3} \times \mathbb{Z}_{3} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{10}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"399",
"N",
"3",
"N",
"0",
"Y",
"10",
"N"
] | [
"NV",
"EX",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following rings, determine its characteristic and determine if it is an integral domain.
$\begin{array}{ccc}\hline & characteristic & integral domain? (Y/N) \\ \hline \mathbb{Z} \times \mathbb{Z}_{37} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{34} \times \mathbb{Z}_{14} & [ANS] & [ANS] \\ \hline \mathbb{Z}_{13} \times \mathbb{Z}_{13} & [ANS] & [ANS] \\ \hline \mathbb{Z}[x] & [ANS] & [ANS] \\ \hline \mathbb{Z}_{18}[x] & [ANS] & [ANS] \\ \hline \end{array}$ | [
"0",
"N",
"238",
"N",
"13",
"N",
"0",
"Y",
"18",
"N"
] | [
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF"
] | [
[],
[],
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0034 | Abstract_algebra | Rings | Units and zero divisors | 2 | [
"inverse",
"zero-divisors"
] | For each of the following elements of the ring $\mathbb{Z}_{45}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 25 & [ANS] & [ANS] \\ \hline 31 & [ANS] & [ANS] \\ \hline 14 & [ANS] & [ANS] \\ \hline 15 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"Y",
"0",
"N",
"16",
"N",
"29",
"Y",
"0"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following elements of the ring $\mathbb{Z}_{30}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 9 & [ANS] & [ANS] \\ \hline 13 & [ANS] & [ANS] \\ \hline 5 & [ANS] & [ANS] \\ \hline 23 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"Y",
"0",
"N",
"7",
"Y",
"0",
"N",
"17"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] | For each of the following elements of the ring $\mathbb{Z}_{18}$, determine if it satisfies the following properties. If it is a unit, enter its inverse in the last column; if it is not, enter 0.
$\begin{array}{ccc}\hline & zero-divisor? (Y/N) & inverse \\ \hline 5 & [ANS] & [ANS] \\ \hline 11 & [ANS] & [ANS] \\ \hline 6 & [ANS] & [ANS] \\ \hline 15 & [ANS] & [ANS] \\ \hline \end{array}$ | [
"N",
"11",
"N",
"5",
"Y",
"0",
"Y",
"0"
] | [
"TF",
"NV",
"TF",
"NV",
"TF",
"NV",
"TF",
"NV"
] | [
[],
[],
[],
[],
[],
[],
[],
[]
] |
Abstract_algebra_0035 | Abstract_algebra | Rings | Units and zero divisors | 3 | [
"zero-divisors"
] | For which of the integers $n \in \lbrace 34, 59, 73, 25, 30, 50, 58, 38 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $45x=45 y$ then $x=y$? [ANS] | [
"(73, 34, 58, 59, 38)"
] | [
"UOL"
] | [
[]
] | For which of the integers $n \in \lbrace 37, 35, 34, 19, 32, 18, 7, 95 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $30x=30 y$ then $x=y$? [ANS] | [
"(7, 19, 37)"
] | [
"UOL"
] | [
[]
] | For which of the integers $n \in \lbrace 87, 92, 35, 82, 21, 29, 55, 19 \rbrace$ is it true that if $x, y \in \mathbb{Z}_{n}$ satisfy $18x=18 y$ then $x=y$? [ANS] | [
"(55, 35, 19, 29)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0036 | Abstract_algebra | Rings | Units and zero divisors | 6 | [
"zero divisors",
"Chinese remainder theorem"
] | Find all integers $n$ in the set $\lbrace 33, 31, 48, 29, 24, 23, 26, 53, 42 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(31, 53, 23, 29)"
] | [
"UOL"
] | [
[]
] | Find all integers $n$ in the set $\lbrace 26, 38, 41, 43, 48, 31, 42 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(41, 31, 43)"
] | [
"UOL"
] | [
[]
] | Find all integers $n$ in the set $\lbrace 37, 42, 26, 24, 38, 23, 53, 28, 43, 47, 33 \rbrace$ such that the ring $\mathbb{Z}_{n}$ satisifies the following properties:
(a) $(xy=0) \Rightarrow (x=0 \textrm{or} y=0)$
(b) $(xy=xz \textrm{and} x \neq 0) \Rightarrow (y=z)$
(c) $(x^2=x) \Rightarrow (x=0 \textrm{or} x=1)$ [ANS]
Reminder: In the ring $\mathbb{Z}_{n}$, $0$ means $0 \pmod{n}$, and equality means congruent modulo $n$. Hint for (c): The Chinese reminder theorem could be useful here. | [
"(37, 47, 43, 23, 53)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0037 | Abstract_algebra | Rings | Ideals and homomorphisms | 6 | [
"ring homomorphisms"
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{45} \rightarrow \mathbb{Z}_{45}$: [ANS]
(b) $\mathbb{Z}_{33} \rightarrow \mathbb{Z}_{11}$: [ANS]
(c) $\mathbb{Z}_{93} \rightarrow \mathbb{Z}_{16}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"45",
"11",
"0"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{23} \rightarrow \mathbb{Z}_{23}$: [ANS]
(b) $\mathbb{Z}_{55} \rightarrow \mathbb{Z}_{8}$: [ANS]
(c) $\mathbb{Z}_{72} \rightarrow \mathbb{Z}_{24}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"23",
"0",
"24"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | Determine the number of possible ring homomorphisms for each pair of rings:
(a) $\mathbb{Z}_{35} \rightarrow \mathbb{Z}_{35}$: [ANS]
(b) $\mathbb{Z}_{139} \rightarrow \mathbb{Z}_{21}$: [ANS]
(c) $\mathbb{Z}_{36} \rightarrow \mathbb{Z}_{9}$: [ANS]
Note: For this problem we do NOT require ring homomorphisms to take the multiplicative identity to the multiplicative identity. | [
"35",
"0",
"9"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0038 | Abstract_algebra | Rings | Ideals and homomorphisms | 2 | [
"ideals"
] | In the ring $\mathbb{Z}_{88}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 88$.
(a) $(51)+(55)$ $($ [ANS] $)$
(b) $(51)(55)$ $($ [ANS] $)$
(c) $(51) \cap (55)$ $($ [ANS] $)$ | [
"1",
"77",
"77"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | In the ring $\mathbb{Z}_{54}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 54$.
(a) $(49)+(9)$ $($ [ANS] $)$
(b) $(49)(9)$ $($ [ANS] $)$
(c) $(49) \cap (9)$ $($ [ANS] $)$ | [
"1",
"9",
"9"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] | In the ring $\mathbb{Z}_{65}$, express each of the following ideals in the form $(m)$ for some element $m$ in the ring, where $0 \leq m < 65$.
(a) $(39)+(18)$ $($ [ANS] $)$
(b) $(39)(18)$ $($ [ANS] $)$
(c) $(39) \cap (18)$ $($ [ANS] $)$ | [
"3",
"52",
"52"
] | [
"NV",
"NV",
"NV"
] | [
[],
[],
[]
] |
Abstract_algebra_0039 | Abstract_algebra | Rings | Ideals and homomorphisms | 3 | [
"maximal ideals"
] | It is a fact that every ideal of $\mathbb{Z}_{144}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{144}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{144}$ containing the ideal $(64)$. Enter a generator for each of these ideals. That is, if you think $(64)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"2"
] | [
"NV"
] | [
[]
] | It is a fact that every ideal of $\mathbb{Z}_{42}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{42}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{42}$ containing the ideal $(36)$. Enter a generator for each of these ideals. That is, if you think $(36)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"(2, 3)"
] | [
"UOL"
] | [
[]
] | It is a fact that every ideal of $\mathbb{Z}_{70}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{70}$.
(a) Determine all maximal ideals of $\mathbb{Z}_{70}$ containing the ideal $(50)$. Enter a generator for each of these ideals. That is, if you think $(50)$ is contained in the maximal ideals $(a)$ and $(b)$, enter $a, b$. [ANS] | [
"(2, 5)"
] | [
"UOL"
] | [
[]
] |
Abstract_algebra_0040 | Abstract_algebra | Rings | Ideals and homomorphisms | 3 | [
"ideals",
"generators"
] | It is a fact that every ideal of $\mathbb{Z}_{72}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{72}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{72}$ that are contained in the ideal $(162)$: $(162) \subseteq I \subseteq \mathbb{Z}_{72}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{72}$ that contain the ideal $(162)$: J \subseteq (162) \subseteq \mathbb{Z}_{72}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 18, 36, 54)",
"(1, 2, 3, 6, 9, 18)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | It is a fact that every ideal of $\mathbb{Z}_{40}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{40}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{40}$ that are contained in the ideal $(56)$: $(56) \subseteq I \subseteq \mathbb{Z}_{40}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{40}$ that contain the ideal $(56)$: J \subseteq (56) \subseteq \mathbb{Z}_{40}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 8, 16, 24, 32)",
"(1, 2, 4, 8)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | It is a fact that every ideal of $\mathbb{Z}_{108}$ is of the form $(b)$ for some element $b$ of $\mathbb{Z}_{108}$.
(a) Find all the ideals $I$ of $\mathbb{Z}_{108}$ that are contained in the ideal $(60)$: $(60) \subseteq I \subseteq \mathbb{Z}_{108}$. In the answer blank below list one generator for each ideal. Separate the generators by commas. [ANS]
(b) Find all the ideals $J$ of $\mathbb{Z}_{108}$ that contain the ideal $(60)$: J \subseteq (60) \subseteq \mathbb{Z}_{108}. As in part (a), list one generator for each ideal, separated by commas. [ANS]
Remember that an ideal contains, and is contained in, itself! | [
"(0, 12, 24, 36, 48, 60, 72, 84, 96)",
"(1, 2, 3, 4, 6, 12)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
Abstract_algebra_0041 | Abstract_algebra | Rings | Ideals and homomorphisms | 2 | [
"ideals"
] | (a) Determine all elements in the ideal $(10)$ of $\mathbb{Z}_{30}$. [ANS]
(b) Determine all elements in the ideal $(10)+(12)$ of $\mathbb{Z}_{30}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{30}$ such that $(10)+(m)$ is a proper ideal of $\mathbb{Z}_{30}$. [ANS] | [
"(0, 10, 20)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28)",
"(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 28)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] | (a) Determine all elements in the ideal $(14)$ of $\mathbb{Z}_{28}$. [ANS]
(b) Determine all elements in the ideal $(14)+(12)$ of $\mathbb{Z}_{28}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{28}$ such that $(14)+(m)$ is a proper ideal of $\mathbb{Z}_{28}$. [ANS] | [
"(0, 14)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26)",
"(0, 2, 4, 6, 7, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] | (a) Determine all elements in the ideal $(10)$ of $\mathbb{Z}_{20}$. [ANS]
(b) Determine all elements in the ideal $(10)+(8)$ of $\mathbb{Z}_{20}$. [ANS]
(c) Determine all elements $m$ of $\mathbb{Z}_{20}$ such that $(10)+(m)$ is a proper ideal of $\mathbb{Z}_{20}$. [ANS] | [
"(0, 10)",
"(0, 2, 4, 6, 8, 10, 12, 14, 16, 18)",
"(0, 2, 4, 5, 6, 8, 10, 12, 14, 15, 16, 18)"
] | [
"UOL",
"UOL",
"UOL"
] | [
[],
[],
[]
] |
Abstract_algebra_0042 | Abstract_algebra | Rings | Quotient rings and polynomial rings | 5 | [
"quotient rings",
"polynomial rings"
] | Find all elements $b \in \mathbb{Z}_{7}$ such that the quotient ring
$\mathbb{Z}_{7} [x]/(x^2+4*x+b)$ is a field. [ANS] | [
"(1, 5, 6)"
] | [
"UOL"
] | [
[]
] | Find all elements $b \in \mathbb{Z}_{2}$ such that the quotient ring
$\mathbb{Z}_{2} [x]/(x^2+x+b)$ is a field. [ANS] | [
"1"
] | [
"NV"
] | [
[]
] | Find all elements $b \in \mathbb{Z}_{3}$ such that the quotient ring
$\mathbb{Z}_{3} [x]/(x^2+x+b)$ is a field. [ANS] | [
"2"
] | [
"NV"
] | [
[]
] |
Abstract_algebra_0043 | Abstract_algebra | Rings | Quotient rings and polynomial rings | 3 | [
"polynomials rings",
"associates"
] | (a) Find all associates of $(7*x^4+7*x^3+8)$ in $\mathbb{Z}_{12} [x]$. Make sure the coefficients are $\geq 0$ and $< 12$. [ANS]
(b) Find all associates of $(1+i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(7*x^4+7*x^3+8, 11*x^4+11*x^3+4, x^4+x^3+8, 5*x^4+5*x^3+4)",
"(1+i, -1-i, -1+i, 1-i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all associates of $(5*x^9+x^3+2)$ in $\mathbb{Z}_{6} [x]$. Make sure the coefficients are $\geq 0$ and $< 6$. [ANS]
(b) Find all associates of $(-6-3i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(5*x^9+x^3+2, x^9+5*x^3+4)",
"(-6-3i, 6+3i, 3-6i, -3+6i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] | (a) Find all associates of $(5*x^9+3*x^7+6)$ in $\mathbb{Z}_{8} [x]$. Make sure the coefficients are $\geq 0$ and $< 8$. [ANS]
(b) Find all associates of $(-6-4i)$ in $\mathbb{Z}[i]$. [ANS] | [
"(5*x^9+3*x^7+6, 7*x^9+x^7+2, x^9+7*x^7+6, 3*x^9+5*x^7+2)",
"(-6-4i, 6+4i, 4-6i, -4+6i)"
] | [
"UOL",
"UOL"
] | [
[],
[]
] |
UGMathBench: A Diverse and Dynamic Benchmark for Undergraduate-Level Mathematical Reasoning with Large Language Models
UGMathBench is a diverse and dynamic benchmark specifically designed for evaluating undergraduate-level mathematical reasoning with LLMs. UGMathBench comprises 5,062 problems across 16 subjects and 111 topics, featuring 10 distinct answer types. Each problem includes three randomized versions.
Paper: https://huggingface.co./papers/2501.13766 GitHub page: https://github.com/YangLabHKUST/UGMathBench.git
An Example to load the data
from datasets import load_dataset
dataset=load_dataset("UGMathBench/ugmathbench", "Trigonometry", split="test")
print(dataset[0])
More details on loading and using the data are on our GitHub page.
If you do find our code helpful or use our benchmark dataset, please cite our paper.
@article{xu2025ugmathbench,
title={UGMathBench: A Diverse and Dynamic Benchmark for Undergraduate-Level Mathematical Reasoning with Large Language Models},
author={Xu, Xin and Zhang, Jiaxin and Chen, Tianhao and Chao, Zitong and Hu, Jishan and Yang, Can},
journal={arXiv preprint arXiv:2501.13766},
year={2025}
}
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