Tagged Questions

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.

76 views

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 ...
41 views

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 ...
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 $\Bbb F_p[x]=\Bbb Z[x]/p = (\Bbb Z/p\Bbb Z)[x]$)
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?
93 views

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: ...
83 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 ...
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 ...
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.
21 views

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 ...
45 views

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..
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 ...
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 ...
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?
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 ...
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 $x\in M$, ...
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 ...
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 ...
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, ...
118 views

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 $X = \{ (x,y) : x^3=y^2, (x,y) \... 0answers 45 views Prime ideal of$\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)$is not a prime ideal. ... 2answers 46 views What are the conditions needed for two principal ideals of a ring to be isomorphic? Given a commutative ring$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$... 3answers 113 views Infinitely many primes$\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$4$... 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? 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 ... 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$5!$... 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 ... 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$are two subgroups of finite ... 1answer 50 views Quotient ring of Gaussian integers$\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$? — For ... 0answers 26 views When is$\{ 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 $$\mathcal{B}\ := \ \{ X^{mk} \ : \ 0 \leq k \leq n-1\... 1answer 36 views Proving that a prime ideal p \subset R yields a prime ideal p[x] \subset R[x] I'm curious as to whether I can have my proof critiqued. Proposition : Let \mathfrak{p} be a prime ideal in a ring R. Show that \mathfrak{p}[x] is a prime ideal in R[x]. Proof : Suppose \... 1answer 22 views Polynomial ring as direct sum of modules I'm aware that this topic has been posted previously. However, I'm not wanting to prove anything. I just want to understand, somewhat intuitively, why R[x] as a module over R is given by$$R[x] =\... 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})=...
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 ...
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 $A$-...