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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The square roots of different primes are linearly independent over the field of rationals

I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using ...
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Units and Nilpotents

If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit. I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) ...
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Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$?

Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$ ? $$\mathbf{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} \mid a,b,c,d\in\mathbf{Q}\}$$ ...
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Comaximal ideals in a commutative ring

Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r ...
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How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to ...
57
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1answer
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Is Lagrange's theorem the most basic result in finite group theory?

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ...
21
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Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. ...
17
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If $G/Z(G)$ is cyclic, then $G$ is abelian

Continuing my work through Dummit & Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in this related MSE answer): Prove that if $G/Z(G)$ is ...
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The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
30
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Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was ...
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Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem. If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
21
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2answers
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$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing I know which connects a group ...
37
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6answers
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Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
14
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4answers
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Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the ...
17
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$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$

I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold. This is exercise 2.11 ...
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An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
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Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots ...
27
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Similar matrices and field extensions

Given a field $F$ and a subfield $K$ of $F$. Let $A$, $B$ be $n\times n$ matrices such that all the entries of $A$ and $B$ are in $K$. Is it true that if $A$ is similar to $B$ in $F^{n\times n}$ ...
14
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Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that ...
35
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2answers
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Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$

Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same ...
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What are the left and right ideals of matrix ring? How about the two sided ideals?

What are the left and right ideals of matrix ring? How about the two sided ideals?
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Irreducible polynomial which is reducible modulo every prime

How to show that $x^4+1$ is irreducible in $\mathbb Z[x]$ but it is reducible modulo every prime $p$? For example I know that $x^4+1=(x+1)^4\bmod 2$. Also $\bmod 3$ we have that $0,1,2$ are not ...
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Prove that if $g^2=e$ for all g in G then G is Abelian.

This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it. (please note that $e$ in the question is the ...
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14answers
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Requesting abstract algebra book recommendations [closed]

I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus. I was ...
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Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
16
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4answers
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A ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
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Group of even order contains an element of order 2

I am working on the following problem from group theory: If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd. That is, for some integer $k$, there are ...
25
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Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is ...
4
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How to prove that $z\gcd(a,b)=\gcd(za,zb)$

I need to prove that $z\gcd(a,b)=\gcd(za,zb)$. I tried a lot, for example, looking at set of common divisors of the two sides, but I can't conclude anything from that. Can you please give me ...
5
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How many irreducible polynomials of degree $n$ exist over $\mathbb{F}_p$?

I know that for every $n\in\mathbb{N}$, $n\ge 1$, there exists $p(x)\in\mathbb{F}_p[x]$ s.t. $\deg p(x)=n$ and $p(x)$ is irreducible over $\mathbb{F}_p$. I am interested in counting how many such ...
10
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A vector space over $R$ is not a countable union of proper subspaces

I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper ...
12
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1answer
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$\operatorname{Aut}(V)$ is isomorphic to $S_3$

I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them here). The lectures come complete with notes and homework problems. Of ...
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Good abstract algebra books for self study

Last semester I picked up an algebra course at my university, which unfortunately was scheduled during my exams of my major (I'm a computer science major). So I had to self study the material, ...
34
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Why can't the Polynomial Ring be a Field?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This ...
11
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1answer
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Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so ...
11
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Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?

In this post we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question: Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? In know these as $\mathbb{Q}$-Vector ...
9
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Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.

Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic. What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
2
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1answer
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If the order divides a prime P then the order is P (or 1)

I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. ...
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Ring Inside an Algebraic Field Extension [duplicate]

Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
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Why is the ring of matrices over a field simple?

Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...
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Why are two permutations conjugate iff they have the same cycle structure?

I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
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Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
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If a group is the union of two subgroups, is one subgroup the group itself?

"Let $G$ be a group, and suppose $G=H \cup K$, where $H$ and $K$ are subgroups. Show that either $H=G$ or $K=G$." Let $h \in H$ and $k \in K$. Then $hk \in H$ or $hk \in K$ (since every element of ...
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Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$

I can't crack this one. Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$ (the only constraint is that $a,b,c,z \in \mathbb{Z}$).
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Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb ...
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2answers
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Example of infinite field of characteristic $p\neq 0$

Can you give me an example of infinite field of characteristic $p\neq0$? Thanks.
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Normal subgroups of $S_4$

Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$? In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?
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The characteristic and minimal polynomial of a companion matrix

The companion matrix of a monic polynomial in 1 variable over a field $F$ plays an important role in understanding the structure of finite dimensional $F[x]$-modules. It is an important fact that the ...
11
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Product of two power series

Say if I define a power series over some arbitrary field $F$ as $$a = \sum^{ \infty }_{i = 0} a_{i} X^{i} $$ Then can I say: $$ab = \sum^{ \infty }_{i = 0} \sum^{ \infty }_{j = 0} a_{i} b_{j} X^{i ...
5
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3answers
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Irreducibility of $X^{p-1} + \cdots + X+1$

Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? Our professor gave us already one, namely to substitute $X$ with ...