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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Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors?

Can we find the GCD of two polynomials in $\mathbb Q[x]$ by representing the coefficients as vectors? For example: $f=x^5+3x^4+x^3+4x^2+1$, and $g=x^5+3x^4+4x^3+3x+1$ Can we represent these ...
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Automorphisms of Abelian groups

Let $A$ be a free Abelian group and $N$ a characteristic subgroup of $A$ such that $A/N$ is finite. I also know that $Aut(A/N)$ and $Aut(N)$ are both finite. I have to prove that $Aut(A)$ is finite. ...
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Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$

I am working on this problem with lots of nesting definitions: Show that $G/Z(R(G))$ is isomorphic to a subgroup of $Aut(R(G))$. For your info, $R(G)$ is called the Radical of $G$, defined as ...
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Define cosine naturally

I'd like to make up a definition of cosine with certain properties of "naturalness". Here is a sketch of the line I wish to follow and I'm hoping someone can state this in a mathematically formal way ...
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Checking if a quotient of a polynomial ring is a field

Show that $\mathbb{F}_2[X]/(x^3+x+1)$ is a field, while $\mathbb{F}_3[X]/(x^3+x+1)$ is not. If I'm right I just need to show that $x^3+x+1$ is irreducible in $\mathbb{F}_2$ and reducible in ...
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Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
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$n_p(GL_2(\mathbb{F}_p))=p+1$

I'm interested in the following problem from Dummit & Foote's Abstract Algebra text (Exercise 40 of Section 4.5): Prove that the number of Sylow p-subgroups of $GL_2(\mathbb{F}_p)$ is $p+1$. ...
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Is there a recurrence formula for finding primitive polynomials in order to construct $\mathbb{F}_{p^n}$?

(In a previous question, I asked some related question but it was very disorientingly formulated. I apologize for that.) Let $p$ be a prime and $n\geq 1$ an integer. The finite field ...
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Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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Find a complex number for the splitting field

Let $E$ the splitting field of $x^3-2 \in \mathbb{Q}[x]$. Applying the algorithm of the proof of the Primitive element theorem, find a complex $c$ with $E=\mathbb{Q}(c)$. $$$$ I have done the ...
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Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification)

Show that a finite group G generated by two elements of order 2 is isomorphic to a dihedral group $D_{2n}$ for some n. (Proof Verification) Proof: Let G be generated by c, b, where $c^2 = b^2 = 1$. ...
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Prove $\sqrt{-7} \not\in \mathbb{Z}\left[\frac{2+3\sqrt{-7}}{4}\right]$

I have the following problem. Consider the ring $\mathbb{Z}$ and define: $$x = \sqrt{-7}\qquad z = \frac{2+3x}{4}$$ Show that $\mathbb{Z}[x] \not\subset \mathbb{Z}[z]$ and $\mathbb{Z}[z] ...
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Prove that an equation has no elementary solution

There are methods proving that a polynomial isn't solvable in radical extensions (see Abel–Ruffini theorem) or proving that an integral or a differential equation has no solutions expressible through ...
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When do Leibniz-like rules lead to unique linear operators

Background Usually one defines differentiation in terms of limits, and then shows that differentiation satisfies the Leibniz (product) rule, $$\frac{d}{dx}(f \cdot g) = f\frac{dg}{dx} + g ...
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Why are centers, centralizers and normalizers called that way?

I know what they are, but where do the names come from?
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Condition on a field that makes every subring an integrally closed domain

I want to know what condition would need to be additionally imposed on a field to make every subring of the field an integrally closed domain.
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Ring of rational power series

Let $A$ be any commutative ring with 1. A power series $f\in A[[t]]$ is called rational if we can find a $g\in A[t]$ such that $fg\in A[t]$. It is clear that the set of rational power series forms a ...
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Consistency of different ways to define the tensor product

It seems my trouble with understanding tensors stems from the following statement: More specifically, the statement: Namely, given $B: V \times W \to U$ and $\xi: U \to \mathbb{R}$, $\xi ...
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co and contravariant vectors, their difference and properties

Very often when talking about covectors, co- and contravariant stuff, it's mentioned that there is no difference in "normal" linear algebra. That the difference only comes "when dealing with curved ...
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When does $n$-dimensional algebra have $m$-dimensional faithful representation?

Suppose we have an $n$-dimensional associative unital algebra $A$ over a field $k$ (assume $\operatorname{char}(k)=0$ and maybe even $k$ is closed). I would like to know what is the minimal ...
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Tensor product of $A_n$ modules/ localisation at ring of differentials

I'm working through Coutinho's "A Primer of Algebraic D-Modules" and I've gotten stuck on the following question: Let $p \in K[x_1, \ldots ,x_n]$ be non-zero, and let $A_n$ be the Weyl Algebra. Show ...
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Existence of a coproduct and representability of a functor

I've found this claim: Let $\mathcal{C}$ be a category; the family $\lbrace C_i \rbrace_{i \in I }$ has a coproduct in $\mathcal{C}$ if and only if the functor $$F : \mathcal{C} \to Set$$ $$A ...
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Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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Three-Dimensional Heisenberg group

Let $H$ be the three dimensional Heisenberg group of quantum mechanics Describe all the conjugacy classes in $H$ as subsets of $\Bbb{R}^3$. Attempt: I was able to find the inner automorphism of the ...
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Does this set can say more about group?

Let $G$ be a finite group, set $\Omega=\{H\leq G\mid N_G(H)=H\}$ It is easy to observe followings; $|\Omega|=1$ if and only if $G$ is nilpotent as normalizer grows in nilpotent groups. For $H\in ...
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Subgroups of Symmetric Group $S_4$ and Isomorphism

During my Algebra class we were given this exercise to solve at home, but I couldn't find any solution and I also did not really get the one our teacher gave us. So, the text was: Given G = S4 = ...
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Abelization of symmetric groups and its subgroups of bounded support

For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the ...
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~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
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Minimal $n$ for embedding of an abelian group into a permutation group $S_n$

Given a finite abelian group $G$, is there a formula or quick algorithm to determine the minimum $N$ so that $G$ can be embedded into the permutation group $S_N$? If $G$ is cyclic of order $n$ I ...
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Unicity inner automorphism symmetric group

I need to prove that the center of the symmetric group $S_n$ with $n\geq 3$ is trivial. I chose to use Lagrange's theorem: bearing in mind that the center is a subgroup, it is easily found that ...
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Number field theoretical analogue of Riemann's Existence Theorem

This question is related to Riemann's Existence Theorem as cited in Neukirch, Schmidt et. al.: Cohomology of Number fields (10.5.1). Let $k$ be a number field, $T \supseteq S \subseteq S_p \cup ...
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Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
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Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
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Extension of Euclidean Domain in which irreducibles have minimal norm

For the ring of polynomials $F[x]$ over a field $F$, there exists a larger ring $\bar{F}[x]$, the ring of polynomials over the closure of $F$, in which irreducibles are linear polynomials -- that is, ...
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$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
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galois extension of imaginary quadratic field

Let $K$ be an imaginary quadratic field, and let $K \subset L$ be a Galois extension. Let $\tau$ denote complex conjugation. Show that $L$ is Galois over $\mathbb{Q}$ if and only if $\tau(L)=L$. My ...
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How to prove the ring $R_J$ is Noetherian without use the theorem of I. S. Cohen?

First define $R:=k\left[{\{X_i\}}_{i\in\mathbb{N}}\right]$ where $k$ is a field (it could be an integral domain as $\mathbb{Z}$ too for example). This ring is an integral domain and it is not ...
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Suppose $G=\{a_1,\dots a_n\}$ is a finite abelian group and $a^2\neq e$ for all nonidentity elements. What is the product $a_1a_2\dots a_n$?

Let $G$ be a finite abelian group such that for all $a \in G$, $a\ne e$, we have $a^2\ne e$. If $a_1, a_2, \ldots, a_n$ are all the elements of $G$ with no repetitions, what is the product $a_1 a_2 ...
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Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto ...
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Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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Clarifying Semi-Direct Products: Example

I'm working through some questions on semi-direct products, and although I can work out these problems (for the most part), I usually have trouble completing them. I have identified some of the things ...
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List of Primes in UFD

Are there websites/databases containing lists ordered by norm of prime/irreducible elements in domains like $\mathbb{Z}[\sqrt{-2}]$ and $\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ for easy ...
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Order of the Normalizer of a Sylow $p$-subgroup in $S_{p}$

Question: Let $p$ be a prime and $P \leq S_{p}$ with $|P| = p$. Prove that $|N_{S_{p}}(P)| = p(p-1)$. I have already solved this problem, but I have come across a proof that is much more elegant than ...
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Highest weights of irreducible components of tensor product of irreducible sl(3)-module.

I am study the representation theory of $sl(3)$ and I have a question about the tensor representation of irreducible $sl(3)$-modules as follows: For each weight $\mu$, let $L(\mu)$ be the irreducible ...
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Application of Sylow Theorems.

I have a group theory exam coming up so I was reviewing through my textbook and I stumbled across a problem that looked interesting: Let $p < q$ be two primes and $G$ be a group of order $pq^3$. ...
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Ultraproduct of the algebraic closure of finite fields

Ok, I have done a little research on the next problem, and there are simple proofs of it using model theory, logic and the Łoś's theorem. But here, the idea is to prove it using only Field Theory, ...
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Every finite map is surjective

I'm trying to understand the proof of the theorem which states that every finite map is surjective in Shafarevich's book: I didn't understand why the part underlined in red is true. I need a ...
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Quadratic equation in $\mathbb{Z}/n\mathbb{Z}$?

I would like to ask for some help about the following problem. Given is a polynomial $\,f(x)=ax^{2}+bx+c\,$ in $\,\mathbb{Z}/ n\mathbb{Z},\,$ we know that this quadratic equation $\,f(x)=0\,$ has ...