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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If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
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3answers
41 views

Subgroup that generates $\mathbb{Z}$

For reference, the example in question is taken from Contemporary Abstract Algebra (Gallian): $$\left< 8, 13 \right> = \mathbb{Z}$$ My first question is to confirm that this is saying that $...
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2answers
48 views

How to prove a given subset is a subgroup [on hold]

If $A, B$ are additive subgroups of a ring $(R,+,\cdot)$, then prove that the set $AB=\{r\in R:r=\sum_{i=1}^n a_i b_i \textrm{ for }a_i\in A,b_i\in B\}$ is an additive subgroup of $R$
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1answer
26 views

$\left(\frac{3}{p} \right)=1$ iff $p\equiv 1\pmod{12}$ or $p\equiv -1\pmod{12}$

Let $p\geq 5$ be a prime. $\left(\frac{3}{p} \right)=1$ iff $p\equiv 1\pmod{12}$ or $p\equiv -1\pmod{12}$. So $\left(\frac{3}{p} \right)=\left(\frac{p}{3} \right)\cdot (-1)^{(p-1)/2}$ and this ...
6
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0answers
86 views

Is an algebraic field extension $k \subseteq K$ normal if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$?

Over a perfect field $k$ it is well known that an algebraic field extension $k \subseteq K$ is normal if and only if $\mathrm{Aut}_{K}(\bar{k})$ is a normal subgroup of $\mathrm{Aut}_{k}(\bar{k})$, as ...
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1answer
17 views

Branch points and Ramification points of a meromorphic map between Riemann Surfaces

Let be $f(z)=\frac{z^3}{(1-z^2)}$ be considered as a meromorphic function on the Riemann Sphere $\mathbb C_{\infty}.$ Consider the affiliated holomoprhic map $F:\mathbb C_{\infty}\rightarrow \mathbb ...
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1answer
29 views

projection of plane on a line

In basic algebra by Nathan Jacobson, he said that a plane can be projected to a line. The text states that "if the plane is the domain and a line is co domain,then one maps any point P in the plane on ...
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0answers
23 views

Why $\Bbb Z[x]/\langle p,f(x)\rangle=\Bbb F_p[x]/\langle \bar f(x)\rangle$?

I don't understand $\Bbb Z[x]/\langle p,f(x)\rangle=\Bbb F_p[x]/\langle \bar f(x)\rangle$. Is it from any theorem? (I know that $\Bbb F_p[x]=\Bbb Z[x]/p = (\Bbb Z/p\Bbb Z)[x]$)
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1answer
45 views

When is $(\Bbb Z/n\Bbb Z)^\times$ cyclic? [duplicate]

Is the group of units $(\Bbb Z/n\Bbb Z)^\times$ always cyclic? Do we need that $n$ is a prime or something?
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1answer
93 views

Why ${S_{n-1}}$ is not a subgroup of ${S_n}$

I know that in general this is not true. Also, I know that subgroups of index n of ${S_n}$ are isomorphic to ${S_{n-1}}$ but why they are not subgroups? For example, why ${S_6}$ is not a subgroup of ${...
19
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1answer
230 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
0
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1answer
20 views

$\Bbb Z[X]/(p, f(X))$ is integral domain

$\Bbb Z[X]/(p, f(X))$ is integral domain. ($p$ is a prime number and $f(X)$ is an irreducible element of $\Bbb Z[X]$) I want to show that if $a, b\notin (p, f(X)), ab\notin(p,f(X))$, but I don't know ...
3
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0answers
47 views

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
17 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
26 views

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

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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3answers
98 views

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
32 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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1answer
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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 ...
4
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1answer
43 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
46 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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2answers
45 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
173 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
34 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
22 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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0answers
31 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
40 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
49 views

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

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
46 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?
3
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1answer
59 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
28 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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0answers
36 views

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
38 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
43 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
118 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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0answers
45 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
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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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3answers
113 views

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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1answer
29 views

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
26 views

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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0answers
21 views

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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0answers
31 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 ...
3
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1answer
31 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 ...
4
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1answer
50 views

Quotient ring of Gaussian integers $\mathbb{Z}[i]/(a+bi)$ when $a$ and $b$ are NOT coprime

The isomorphism $\mathbb{Z}[i]/(a+bi) \cong \Bbb Z/(a^2+b^2)\Bbb Z$ is well-known, when the integers $a$ and $b$ are coprime. But what happens when they are not coprime, say $(a,b)=d>1$? — For ...
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0answers
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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
36 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
22 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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0answers
24 views

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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1answer
31 views

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 $A$-...