User Michael Morrow

6 votes

1 answer

314 views

Show $\sqrt[3]{5}$ is not contained in any cyclotomic extension of $\mathbb{Q}$.

asked May 8, 2020 at 20:15

5 votes

1 answer

74 views

Let $|G|=2^np$, $p$ an odd prime, $H\unlhd G$ a Sylow $2$-subgroup with $H\cong(\Bbb{Z}/2\Bbb{Z})^n$, $p\nmid 2^n-1$. Prove $Z(G)$ is nontrivial.

abstract-algebra group-theory normal-subgroups sylow-theory

asked Jun 3, 2020 at 23:53

4 votes

2 answers

194 views

Subspace of polynomials vanishing to order $n$

linear-algebra

asked May 3, 2020 at 0:06

4 votes

0 answers

68 views

Find all ring maps $\mathbb{Q}[x]/(x^{101}+2)\longrightarrow\mathbb{Q}[x]/(x^{501}-2)$

abstract-algebra ring-theory field-theory solution-verification

asked Apr 11, 2020 at 20:36

4 votes

3 answers

90 views

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

abstract-algebra polynomials field-theory finite-fields

asked May 17, 2020 at 3:22

4 votes

1 answer

96 views

Is the space $\text{Map}(S^1,S^1)$ of continuous maps on $S^1$ compact? (Compact-open topology)

general-topology compactness

asked May 29, 2021 at 18:23

3 votes

1 answer

73 views

Any two components of a topological group are homeomorphic

general-topology solution-verification topological-groups

asked Jun 1, 2021 at 22:35

3 votes

1 answer

158 views

Can the Klein bottle cover the torus? Show the Klein bottle can't be covered by a space $X$ with $\pi_1(X)=\mathbb{Z}/3\mathbb{Z}$.

algebraic-topology covering-spaces

asked Jun 3, 2021 at 20:54

3 votes

3 answers

104 views

Subgroups of $C_2\times C_6$

abstract-algebra group-theory

asked May 7, 2020 at 22:54

3 votes

2 answers

86 views

Show $y^3+xy+x^2(x-1)^2\in\mathbb{R}[x,y]$ is irreducible

abstract-algebra polynomials ring-theory irreducible-polynomials

asked Mar 25, 2020 at 18:46

3 votes

1 answer

103 views

Integral of function equals integral of measure - Tonelli's Theorem

real-analysis lebesgue-integral lebesgue-measure

asked Apr 17, 2020 at 3:22

3 votes

1 answer

121 views

Odd-dimensional $\mathbb{R}$-vector space has a one-dimensional $\varphi$-invariant subspace

linear-algebra

asked Jun 3, 2020 at 22:52

3 votes

1 answer

383 views

Prove quotient of graded ring with graded ideal is a graded ring

abstract-algebra ring-theory commutative-algebra graded-rings

asked Jul 30, 2020 at 17:05

2 votes

1 answer

508 views

Minimal Gröbner bases have the same leading terms

abstract-algebra commutative-algebra groebner-basis

asked Sep 16, 2020 at 2:10

2 votes

0 answers

60 views

Classification of monomials in a monomial ideal

abstract-algebra ring-theory monomial-ideals

asked Jul 5, 2020 at 22:38

2 votes

2 answers

102 views

Two questions about graded rings

abstract-algebra ring-theory

asked Jul 9, 2020 at 19:05

2 votes

0 answers

51 views

Your favorite way to think of $k[x_1,\ldots,x_n]$ modulo some graded ideal?

abstract-algebra polynomials ring-theory commutative-algebra soft-question

asked Jul 21, 2020 at 19:19

2 votes

2 answers

80 views

Equivalent properties of graded ideals

abstract-algebra ring-theory commutative-algebra

asked Jul 29, 2020 at 19:48

2 votes

2 answers

293 views

Restriction of surjective linear map is isomorphism

linear-algebra

asked Jun 3, 2020 at 22:15

2 votes

1 answer

170 views

Non-trivial normal subgroup of $G$ intersects the center $Z(G)$ non-trivially

abstract-algebra group-theory normal-subgroups

asked Jun 2, 2020 at 3:07

2 votes

1 answer

35 views

Automorphism of vector space $V$ such that $\varphi(S_1)=S_2$

linear-algebra solution-verification

asked Jun 4, 2020 at 18:05

2 votes

1 answer

413 views

How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$

abstract-algebra field-theory galois-theory cyclotomic-fields

asked May 20, 2020 at 20:33

2 votes

2 answers

47 views

Minimal polynomial of $\varphi:g+(f)\mapsto xg+(f)$ is $f$.

linear-algebra abstract-algebra field-theory

asked May 28, 2020 at 21:39

2 votes

3 answers

79 views

Number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$

abstract-algebra field-theory finite-fields

asked May 30, 2020 at 4:00

2 votes

1 answer

77 views

$K$-invariant $F$-homomorphism over finite normal field extension $K/F$

abstract-algebra field-theory

asked Mar 27, 2020 at 16:20

2 votes

1 answer

123 views

Find a Galois extension $\mathbb{Q}\subset K$ with $\text{Gal}(K/\mathbb{Q})\cong\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$

abstract-algebra field-theory galois-theory

asked Mar 29, 2020 at 18:57

2 votes

1 answer

79 views

Show $(29,x^2+1)\subset\mathbb{Z}[x]$ is not a maximal ideal

abstract-algebra ring-theory ideals alternative-proof

asked Apr 14, 2020 at 2:27

2 votes

1 answer

68 views

Direct sum of subspaces using Cayley-Hamilton

linear-algebra direct-sum cayley-hamilton

asked Dec 6, 2019 at 23:56

2 votes

2 answers

108 views

Show set of matrices forms a finite field of order 27

linear-algebra abstract-algebra matrices field-theory finite-fields

asked Mar 23, 2020 at 21:11

2 votes

1 answer

45 views

How many elements of order $p^7$ are in $C_{p^5}\times C_{p^6}\times C_{p^7} \times C_{p^8}\times C_{p^9}$?

abstract-algebra group-theory

asked May 16, 2020 at 18:44

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92 Questions

Show $\sqrt[3]{5}$ is not contained in any cyclotomic extension of $\mathbb{Q}$.

Let $|G|=2^np$, $p$ an odd prime, $H\unlhd G$ a Sylow $2$-subgroup with $H\cong(\Bbb{Z}/2\Bbb{Z})^n$, $p\nmid 2^n-1$. Prove $Z(G)$ is nontrivial.

Subspace of polynomials vanishing to order $n$

Find all ring maps $\mathbb{Q}[x]/(x^{101}+2)\longrightarrow\mathbb{Q}[x]/(x^{501}-2)$

Roots of $x^{p^{n-1}}+\ldots+x^p+x$ in $\mathbb{F}_{p^n}$

Is the space $\text{Map}(S^1,S^1)$ of continuous maps on $S^1$ compact? (Compact-open topology)

Any two components of a topological group are homeomorphic

Can the Klein bottle cover the torus? Show the Klein bottle can't be covered by a space $X$ with $\pi_1(X)=\mathbb{Z}/3\mathbb{Z}$.

Subgroups of $C_2\times C_6$

Show $y^3+xy+x^2(x-1)^2\in\mathbb{R}[x,y]$ is irreducible

Integral of function equals integral of measure - Tonelli's Theorem

Odd-dimensional $\mathbb{R}$-vector space has a one-dimensional $\varphi$-invariant subspace

Prove quotient of graded ring with graded ideal is a graded ring

Minimal Gröbner bases have the same leading terms

Classification of monomials in a monomial ideal

Two questions about graded rings

Your favorite way to think of $k[x_1,\ldots,x_n]$ modulo some graded ideal?

Equivalent properties of graded ideals

Restriction of surjective linear map is isomorphism

Non-trivial normal subgroup of $G$ intersects the center $Z(G)$ non-trivially

Automorphism of vector space $V$ such that $\varphi(S_1)=S_2$

How to find generators for the subfields of $\mathbb{Q}(\zeta_{12})$

Minimal polynomial of $\varphi:g+(f)\mapsto xg+(f)$ is $f$.

Number of elements $a\in\mathbb{F}_{5^4}$ such that $\mathbb{F}_{5^4}=\mathbb{F}_5(a)$

$K$-invariant $F$-homomorphism over finite normal field extension $K/F$

Find a Galois extension $\mathbb{Q}\subset K$ with $\text{Gal}(K/\mathbb{Q})\cong\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$

Show $(29,x^2+1)\subset\mathbb{Z}[x]$ is not a maximal ideal

Direct sum of subspaces using Cayley-Hamilton

Show set of matrices forms a finite field of order 27

How many elements of order $p^7$ are in $C_{p^5}\times C_{p^6}\times C_{p^7} \times C_{p^8}\times C_{p^9}$?