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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Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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3answers
10 views

Sylow $p$-subgroups in $S_p$ and order $p$ elements

I realise this question has been asked a few times before, but I don't understand the answers. What is the number of Sylow p subgroups in S_p? Why are the order $p$ elements in $S_p$ exactly cycles ...
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1answer
17 views

If $K\leq H\leq G$ (not necessarily finite groups). Then prove that $[G:K]=[G:H]\cdot [H:K]$

Let $K\leq H\leq G$ (not necessarily finite groups). Why do we have $[G:K]=[G:H]\cdot [H:K]$? I can't figure out a proof in the setting of possibly infinite groups and non-normal subgroups.
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Field with $125$ elements

I want to construct a field with $125$ elements. My idea is to consider the polynomial ring $\Bbb F_5[x]$. It is enough to find an irreducible polynomial $f\in \Bbb F_5[x]$ of degree $3$ because then $...
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1answer
16 views

Finding the $\gcd$ of polynomials in $\Bbb R[x]$

Let $f(x)=6x^3-10x^2-6x+10$ and $g(x)=3x^2-14x+15$ in $\Bbb R[x]$. I want to find the $\gcd$ of these two polynomials. I am not really sure how to do this in general, but my approach was as follows: ...
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25 views

Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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1answer
22 views

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?} $$ It almost feels like the nilradical (which is zero for any ...
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1answer
31 views

A prime ideal $\mathfrak{p} \subset \mathcal{O}_K$ lies above/over $p$ if $\mathfrak{p}\cap \mathbb{Z} = p\mathbb{Z}$ [duplicate]

In the concrete case that $\mathcal{O}_K = \mathbb{Z}[i]$, and $\mathfrak{p} = (1+i)$, how to make sense of $\mathfrak{p}\cap \mathbb{Z}$? I want to know if (1+i) lies above/over 2.
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What is number of irreducible characters in modular representation?

In ordinary representation, I know that the number of irreducible characters is the number of conjugacy classes, what about the modular representation? Can I find a good and simple book on modular ...
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32 views

Let $E\supseteq F$ be a splitting field of $f(x)=x^6+1\in F[x]$. Find $[E:F]$.

Let $E\supseteq F$ be a splitting field of $f(x)=x^6+1\in F[x]$. Find $[E:F]$ for $F=\mathbb{Z}_2$ and $F=\mathbb{Q}$. I think in $\mathbb{Z}_2$, we can rewrite it as $f(x)=x^6-1=(x^3-1)(x^3+1)=(x^3-...
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1answer
155 views

Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
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2answers
28 views

$G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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1answer
17 views

$E/F$ is a finite Galois extension. Let $b\in E$, and $b_1=b, b_2…$ are the orbit of b under the action

Let $E/F$ be a finite Galois extension. So $E=F(a)$. Let $b\in E$, and let $b_1=b, b_2,...,b_n$ be the orbit of $b$ under the action of the Galois group $G$. (a)Show that the minimal polynomial of $...
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30 views

Let $G$ be a nilpotent and $H<G$. Is $H$ a Nilpotent?

This is my question: Let $G$ be a nilpotent and $H<G$. Is $H$ a Nilpotent? Can you HELP me with the proof? Thank you so much..
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3answers
34 views

Every alternating permutation is a product of 3-cycles [duplicate]

Show that every element in $A_n$ (alternating group of degree $n$) for $n \ge 3$ can be expressed as a $3$-cycle or a product of three cycles. I understand that if $n$ is odd, then any element can ...
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2answers
44 views

Non-abelian Order of $6$ is isomorphic to $S_3$ [on hold]

I know that it's duplicate, but , How can I prove it? I know that must be element "$a$" of order 2, and element "$b$" of order 3. What is the next step? In fact, from what I search it is claimed that ...
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3answers
44 views

Raising element of field to characteristic power

I came across this in a set of notes. Let $K$ be a field of characteristic $p$ and let $\lambda\in K$. Then $$\lambda^{p-1}=1.$$ I've never seen this before. Is it correct?
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1answer
35 views

How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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0answers
25 views

Is power-associativity an equational property?

A magma $M$ is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as $x^mx^n=x^{m+n}$ for all $m,n$ positive integers and $x\in M$, ...
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Algebra of Endomorphisms of a functor $\mathcal{F}$

Recently, I was reading a paper and stumbled upon something for which I don't find the formal definition unfortunately in any textbook that have in my hands. It's the notion of the endomorphism ...
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2answers
35 views

Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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1answer
39 views

$\Bbb Z[\sqrt{-5}]$ is not a PID [duplicate]

I want to show: In a PID $R$ two elements $a,b\in R$ always have a greatest common divisor. Therefore $\Bbb Z[\sqrt{-5}]$ is not a PID. For the first part: $I=\{ax+by:x,y\in R\}$ is an ideal, ...
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1answer
47 views

Is this regular function globally rational?

Let $\mathbb{k}$ be an algebraically closed field of characteristic not 2 or 3, and let $X \subseteq \mathbb{A}^2_\mathbb{k}$ be the locally closed subset given by $X = \{ (x,y) : x^3=y^2, (x,y) \...
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1answer
29 views

Prime ideal of $\Bbb Z[X]$ is $(0)$ or $(f)$ or $(p, f)$

Prime ideal of $\Bbb Z[X]$ is $(0)$ or $(f)$ or $(p, f)$ ($f\in\Bbb Z[X]$, $f$ is an irreducible element. And $p$ is a prime number) By the statement above, $(x^3+2,2x^2+3)$ is not a prime ideal. ...
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2answers
39 views

What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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Infinitely many primes $\equiv 3 \mod 4$

Question 1 Is the following proof of the infinitude of primes $\equiv 1\mod 4$ okay? Consider a prime divisor $p\mid (n!)^2+1$. Then $(n!)^2\equiv -1 \mod p$, hence $n!$ has multiplicative order $4$ ...
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Is $\Bbb Z[i]$ a Euclidean ring? [duplicate]

Is $\Bbb Z[i]$ a Euclidean ring? If not, what would be the simplest way of seeing that $\Bbb Z[i]$ is a PID?
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1answer
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Given a commutative ring $R$ and a monic polynomial $p(x) \in R[x]$ is $R[x]/\langle p(x) \rangle$ always a finite integral extension of $R$?

I suspect this to be true based on the fact that $p(x)$ is monic, so it should be the case that $R[x]/\langle p(x) \rangle$ is a finitely generated module over $R$, but I have no good reference for ...
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Show that $A_5$ (alternating group of degree $n$) has $24$ elements of order $5$, $20$ elements of order $3$, and $15$ elements of order $2$.

Show that $A_5$ (alternating group of degree $n$) has $24$ elements of order $5$, $20$ elements of order $3$, and $15$ elements of order $2$. I'm a little confused on this. Wouldn't there be $5!$ ...
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28 views

The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
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1answer
30 views

Problem with proof of $H \cap K $ is of finite index if $ H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...
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When is $\{ X^{mk} \ : \ 0 \leq k \leq n-1\} $ a basis for $R[X]/(f)$?

Let $R$ be a commutative ring and $f \in R[X]$ irreducible with degree $n$. Let $m$ be an integer such that $0 \leq m \leq n-1$. Can we say that $$ \mathcal{B}\ := \ \{ X^{mk} \ : \ 0 \leq k \leq n-1\...
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1answer
34 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
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1answer
20 views

Polynomial ring as direct sum of modules

I'm aware that this topic has been posted previously. However, I'm not wanting to prove anything. I just want to understand, somewhat intuitively, why $R[x]$ as a module over $R$ is given by $$R[x] =\...
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Proof verification: Show that the fixed field is $\mathbb{Q}(\sqrt{3})$

Let $H$ be the subgroup $\{i,\alpha\}$ of $\text{Gal}_{\mathbb{Q}}\mathbb{Q}(\sqrt{3},\sqrt{5}),$ where $i$ is the identity map and $\alpha$ is defined as $\alpha(\sqrt{3})=\sqrt{3}$,$\alpha(\sqrt{5})=...
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3answers
27 views

How to represent polynomial rings in multiple variables.

If $R$ is a ring, we can form the polynomial ring in $x$ as $$R[x]=\{\sum_{i=1}^nax^i|a \in R \wedge n\in \mathbb{N} \cup\{0\}\} $$ where the $\sum_{i=1}^nax^i$ are formal sums. Every source I look ...
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The sum of a family of submodules

I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions. Firstly, suppose $M$ is an $A$-...
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1answer
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Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
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2answers
50 views

Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
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3answers
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Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
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1answer
46 views

Not Abelian group G with Z(G) that contains only two elements? [on hold]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
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$R/Rg$ is a field iff $g\in R$ is irreducible.

Let $R$ be a PID and $g\in R$. I want to show: $R/Rg$ is a field iff $g\in R$ is irreducible. I.e. I want to show that all $a\notin Rg$ are invertible modulo $g$ iff $g$ is irreducible. So if I ...
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How to prove in $r_1p_1 +r_2p_2 =u\gcd(p_1,p_2)$, $u$ is a uniformly random polynomial.

Hypothesis: All polynomials are defined over a finite field $\mathbb{F}_p$, where $p$ is a large prime number (e.g. 128-bit prime number). Assume we have two fixed polynomials $p_1$ and $p_2$ of ...
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Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
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2answers
22 views

Order of product multiplication of $n$-cycles that may not commute.

Suppose that $\beta$ is a $10$-cycle. For which integers $i$ between $2$ and $10$ is $\beta^i$ also a $10$-cycle? The question I have is concerning the order of checking the powers of $\beta^i$. ...
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60 views

A test problem about algebraic integers in complex field

In a recent algebraic test, I meet this problem: Let R be the ring of algebraic integers in C, K is the field of algebraic numbers in C. Let a be an element of K such that the ring R[a] is ...
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0answers
26 views

Minimal injective resolutions isomorphism [on hold]

How can I prove that given an $A$-module $M$ two injective resolutions of $M$ are isomorphic as complexes? Thank you, have a nice day Asdrubale
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0answers
17 views

noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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38 views

Koszul complex: isomorphism between $K(a_1,\ldots, a_n;A) \simeq K(a_1;A) \otimes \cdots \otimes K(a_n;A)$

Given $a_1,\dots,a_n\in A$, with $A$ a suitable ring, my algebra teacher defined the Koszul complex associated to $a_1,\dots,a_n$ with coefficients in $A$ in this way: $$K(a_1,\dots,a_n;A):=\...
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1answer
33 views

Minimal graded free resolution of the ideal $(x^3,xy^2,y^5)$

I am looking for a detailed explanation of every step of the construction of a graded free resolution of the ideal $(x^3,xy^2,y^5) \subseteq S=K[x,y]$ where $K$ is an arbitrary field. I saw several ...