Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Let A and B be ideals of a ring and C a prime ideal. Prove if the intersection of A and B is a subset of C then Either A or B is a subset of C

Claim: Let $A$ and $B$ be ideals of a commutative ring $R$ and C a prime ideal of $R$. Suppose that the intersection of $A$ and $B$ is a subset of $C$, prove either $A$ or $B$ is a subset of $C$. My ...
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Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H.

Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H. Show that if K is a normal subgroup of G, then K is a normal subgroup of H.
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24 views

Showing a subgroup contains the identity element

Let $G$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ along with $+$. Show that $H$ defined by $H=\{f:f(x)=0 \text{ for all } x \in [0,1]\}$ is a subgroup. I am able to show $H$ ...
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Some proof about $\mathbb Z[\sqrt{-5}]$

For $\mathbb Z[\sqrt{-5}]$, $6=2\cdot3=(1+\sqrt5)(1+\sqrt{-5})$; if $2=\gamma\cdot \delta$, then $\gamma$ or $\delta$ is a unit, or $|\gamma|=2$ or $3$. The same can be said of $|\delta|$. These ...
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1answer
18 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\theta \colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi ...
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19 views

Subgroups of $G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}$

Let $$G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}.$$ Is this a subgroup? Clearly $id \in G$, but what is known about the order of $\sigma^{-1}$ and of $\sigma \circ \tau$ (where ...
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21 views

Reduction modulo $p$ of $x^4 + 3x^3 -21x^2 -62x -40$ with a multiple root

Let us consider $$g(x) = x^4 + 3x^3 -21x^2 -62x -40 \in \mathbb{Z}[x].$$ How does one find the primes $p>0$ such that the reduction of $g(x)$ modulo $p$ has a multiple root?
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13 views

$K(x)$ not stable relative to $K(x,y)$ and $K$

Prove that in the extension of an infinite field $K$ by $K(x,y)$, the intermediate field $K(x)$ is Galois over K, but not stable (relative to $K(x, y)$ and $K$). I know that if K(x) is algebraic it ...
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23 views

What does Cayley table for a group $(\mathbb{Z}_5^*,\cdot_5)$ tell us?

Firsly, I would like you to explain me what $\mathbb{Z}_5^*$ means. My teacher told me it is a group of units of $\mathbb{Z}_5$, but I'm not sure what a group of units is. If ...
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20 views

Does the order of a finite group divide the product of degrees of a system of parameters of the invariant algebra?

Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ ...
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1answer
16 views

Pick out a polynomial such that ideal $J=q(x)R$ , where $q(x)$ is polynomial and $R$ is ring

In the ring of polynomials $R =\mathbb Z_5[x]$ with coefficients from the field $\mathbb Z_5$, consider the smallest ideal $J$ containing the polynomials, $p_1(x) = x^3 + 4x^2 + 4x + 1$ $p_2(x) = ...
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14 views

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$

Show that $p = u \cdot (\zeta -1)^{p-1}$, where $u$ is an invertible element of $Z[\zeta]$. This outcome is the result of this link. So I think I have to use the previous result and the ...
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21 views

Number of field homomorphisms [on hold]

Let $E$ be a finite field extension over $K$. Show there are at most $[E:K]$ $K$-homomorphisms $E \to F$, where $F$ is an algebraic closed field extension of $E$.
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26 views

Product of two primitive roots $\bmod p$ cannot be a primitive root.

I recently proved that the product of all primitive roots of an odd prime $p$ is $\pm 1$ as an exercise. As a result, I became interested in how few distinct primitive roots need to be multiplied to ...
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10 views

Show that all simple modules of ${M_n}{(D)}$ are isomorphic to $D^n$, where $D$ is a division ring. [duplicate]

I am trying to solve part (c) of the following Representation Theory question: Let $D$ be a division ring and let $n$ be a positive integer. For $ 1 \leq l \leq n $ let $$C_l= \{A = (a_{ij}) \in ...
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28 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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2answers
19 views

Number of solutions for the given equation in finite field of order 32.

Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$. I have figured out that non zero elements of this field forms a cyclic multiplicative group of ...
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1answer
12 views

Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$, where $N_G = {\sum}_{g \in G} {g} \in F[G]$. [duplicate]

I am trying to solve this Representation Theory question: Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a ...
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16 views

Show ring isomorphisms $End_R{({S_1}^{n_1} \oplus \ldots \oplus {S_r}^{n_r} )} \cong End_R{({S_1}^{n_1})} \times \ldots \times End_R{({S_r}^{n_r})}$

I have been struggling with this Representation Theory question for the past week: Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomorphic to $S_j$ whenever $i \neq j$ ...
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48 views

$G$ is a finite group, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$.

Prove / disprove: Let $G$ be a finite group of order $n$, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$. I can't find a group that will satisfy this condition so I think it's ...
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42 views

Finding Galois Group

Let $\mathbb Q\subset\mathbb Q(\sqrt{3}+\sqrt[3]{5})$. I want to find the Galois group of the given field extension. It would be easy for me if I could find a basis of the given field extension ...
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38 views

Variation on Fermat Little Theorem

Does the following variation of Fermat Little Theorem hold? How do you prove it? Let $p$ be a prime number greater than $3$. Then there exist a natural non-prime $m > 1$ such that ...
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7 views

Is it possible to convert a general quintic to Brioschi form in one single transformation?

The standard method of converting a general quintic to Brioschi form $X^5-10CX^3+45C^2X-C^2=0$ proceeds in two steps which required the extraction of a square root. One first converts to the ...
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1answer
30 views

Is this a linear transformation? in the context of group representations

Let $G$ be a group. A regular representation is given as $V=\mathbb{C}[G]$, a vector space, where $l: G \to GL(V)$ be the action is given by $l(g)(\alpha)(h) = \alpha (g^{-1}h)$ for all $g,h\in G, ...
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20 views

$>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$

Let us define $R = k[x_1,\dots,x_t,x_{t+1},\dots,x_n]$; then it can be shown that $>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$ for all $1\leq i \leq t, t+1 \leq j \leq n$ ...
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1answer
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Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
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1answer
15 views

How do you prove a valuation ring is a subring?

Let's say I have a field $\mathbb{F}$. Now suppose I take the set $R = \{x \in \mathbb F^{\times}: \ y(x) \ge 0\} \cup \{0\}$ where $y$ is a function $y:\mathbb F^{\times} \rightarrow \mathbb{Z}$ ...
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1answer
29 views

How many elements are there in $\mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2})$?

Let $\mathbb{Z}[\sqrt{2}]:= \lbrace a+b\sqrt{2}|a,b \in \mathbb{Z} \rbrace$. How many elements are there in $\mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2})$? I know that every equivalence class of ...
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2answers
26 views

Proving that a homomorphism between two rings is surjective

The problem: ($\mathscr{F}(\mathbb{R})$ is the set of real valued functions) Let $\phi:\mathscr{F}(\mathbb{R})\to\mathbb{R}\times\mathbb{R}$ be a function defined by $\phi(f)=(f(0),f(1))$ Prove ...
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33 views

Show that the ideal generated by $x^2-2$ is maximal

Let $A = \mathbb{Q}[x]$. Show that the ideal generated by $x^2-2$ is maximal. I think it is sufficient to show that $\mathbb{Q}[x]/(x^2-2) \cong \mathbb{Q}\sqrt{2}$, where $\mathbb{Q}\sqrt{2} = \{a + ...
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36 views

Show that $(2,1+\sqrt{-5})$ is a maximal ideal in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]

Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$ an ideal generated by $2$ and $1+\sqrt{-5}$. Show that $I$ is a maximal ideal. So I tried to prove that if $a \notin I$ then $(I,a)$ must be ...
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Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$?

Personal question: Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$? I know the ideal in $\mathbb{Q}[x]$ generated by $x^2-2$ is maximal, considering ...
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Groebner basis and prime ideals.

Let $I$ be an ideal in a polynomial ring $K[x,y_1,\dots,y_n]$ and assume that $I \cap K[x]\neq (0)$. Let $>$ be an elimination ordering for $\{y_1, \dots, y_n\}$ and $G$ is a Groebner basis for $I$ ...
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1answer
16 views

Semidirect product when the action factors

Suppose I am forming a semidirect product of finite groups $G \rtimes_\theta A$. Here, $\theta : A \to \mathrm{Aut}(G)$ is a homomorphism. Now, suppose I notice that the homomorphism $\theta$ factors ...
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31 views

Finite number of maximal ideals of bounded norm

Suppose that we have an integral extension of rings $R\subseteq S$ and $S$ is finitely generated as $R$-module or as $R$-algebra, and $R/\mathfrak m$ is finite for all maximal ideals and $S/\mathfrak ...
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The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
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If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit.

If $a \ne b$ in a ring $R$ satisfy $a^3 = b^3$ and $a^2b = b^2a$, show that $a^2 + b^2$ is not a unit. So I am thinking that I should be able to do this by contradiction. So if I assume there is some ...
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Uniqueness of Smith normal form in Z (ring of integers)

It is a very well known fact that Smith Normal Form has proven useful when dealing with the development of the structure theorem of finitely generated abelian groups. In this context, there is an ...
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1answer
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Let $R$ be an integral domain. Every associate of a prime element of $R$ is prime.

Let $c$ be prime and $d$ be an associate of $c$. Claim: $d$ is prime. $c$ prime $\Rightarrow c\neq 0$, nonunit Suppose that $c\vert ab$. Since $c$ is prime, $c\vert a$ or $c\vert b$ $c\sim d ...
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Is a factorable polynomial invertible?

The reason there exists no quintic formula that finds the roots of a quintic polynomial is simply because some quintic polynomials are irreducible. But reducible quintic polynomials may be invertible ...
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If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$.

Let $1_R$ be the identity of a ring $R$ and let $\mathbb{Z}1_R=\{k1_R\mid k\in\mathbb{Z}\}$. If char$R=n$, show that $\mathbb{Z}1_R\cong \mathbb{Z}_n$. So my thought is I just have to think of some ...
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Is $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$?

I'm trying to determine whether $\mathbb{Z}[x]/(x^2 + x + 1, 9)$ isomorphic to $\mathbb{F}_{81}$, but I'm confused since $\mathbb{Z}/9\mathbb{Z}$ is not a field, but $x^2 + x + 1$ is irreducible in ...
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1answer
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Poincare series and the Hilbert polynomial of $A = A_0[X_1,\dots , X_s]$

Let $A = A_0[X_1, \dots , X_s]$ be a polynomial ring in $s$ variables over an Artin ring $A_0$. This is a graded ring, and can be regarded as a graded module over itself. 1. What are the ...
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54 views

Show that the elements of the form $1+\zeta + \zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$

Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$. I know ...
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3answers
27 views

If $S$ and $T$ are subrings of $R$, is $S+T$ a subring of $R$?

If $S$ and $T$ are subrings of $R$, is $S+T=\{s+t\mid s\in S, t\in T\}$ a subring of $R$? So I think that $S+T$ is a subring, but I am getting stuck trying to prove it. Clearly since $S$ and ...
2
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1answer
35 views

Determine the quotient group $S_4$/N i.e. determine it is isomophic to $S_3$ [duplicate]

Let $N = \{1,(12)(34),(13)(24),(14)(23)\}$. Determine the quotient group $S_4$/N i.e. determine it is isomophic to $S_3$ I computed the cosets: $N, (12)N, (13)N, (14)N, (123)N, (234)N$, and the ...
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1answer
57 views

Only two groups of order 10. [duplicate]

I have shown a group of order 10 has a cyclic subgroup of order 5. But now I am stuck. Any hints?
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61 views

Is $S_3$ a normal subgroup in $S_4$? [on hold]

Let $G=S_4$ and $H=S_3$. Is $H$ a normal subgroup in $G$? What is the fastest way to do this problem. I am sure the answer is no. Is it quicker to find an example of a left coset and right coset ...
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1answer
37 views

Elimination Ordering for the ring $k[x,y]$

How to show that the only elimination ordering on the ring $k[x,y]$ is the lexicographic ordering? (Ene and Herzog, Gröbner Bases in Commutative Algebra, Problem 3.1.) Definition (Elimination ...
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3answers
56 views

Is the following set a group?

Let $ G= \begin{pmatrix} a & a\\ a & a\\ \end{pmatrix} $ where $a\in \Bbb R, a \neq0$. I need to show that $G$ is a group under matrix multiplication. The ...