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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Reciprocal-based field axioms

In this question it is shown that being able to compute reciprocals (together with sums and differences) is enough to do do multiplication in a field of characteristic $\ne 2$. That made me wonder: ...
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Examples of Noncommutative Noetherian Rings in which Lasker-Noether Fails?

I'm writing a paper on Emmy Noether for my introductory Abstract Algebra class, and I'm looking for examples of noncommutative Noetherian rings in which the Lasker-Noether theorem fails to hold. ...
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Natural Euclidean Function Not Satisfying the $d$-inequality

Let me provide some background before I begin (although I feel as though it's hardly needed): Let $R$ be an integral domain. I call a function $d:R-\{0\}\to\mathbb{N}\cup\{0\}$ a Euclidean ...
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141 views

Is $\mathrm{Mod}(-)$ a functor?

Note that if we have a ring $R$, we can talk about $R$-modules, and if we have a ring homomorphism $R\to S$, there is a map from $R$-modules to $S$-modules given by $-\otimes_RS$ (just assume ...
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Identifying a certain subgroup of a free group

Let $F$ be the free group on the generators $a_1,\ldots,a_n$. Define homomorphisms $\phi_i:F\to F$ by $\phi_i(a_j)=a_j^{1-\delta_{ij}}$, where $\delta_{ij}$ is the Kronecker delta; basically, $\phi_i$ ...
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292 views

Inverse of the Grothendieck group construction?

Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that ...
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367 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
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81 views

Restriction of trivariate polynomial to $1$ variable

Let $p(x,y,z): \mathbb{F}^3 \to \mathbb{F}$ be a trivariate polynomial of degree $d \ll |\mathbb{F}|$. We choose uniformly at random an affine $1$-dimentional space $\ell = \{(a_1,a_2,a_3)t + (b_1,b_2,...
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175 views

Is there an integral domain with a lot of residue fields of the same characteristic?

Is there a commutative integral domain $R$ in which: every nonzero prime ideal $Q$ is maximal, and for every prime power $q\equiv 3 \bmod 8$, there is a maximal ideal $Q$ of $R$ such that $...
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193 views

A relation between permanents and determinants

I have skimmed this video that I found on mathoverflow: http://tube.sfu-kras.ru/video/407?playlist=397 At about 15:05 the lecturer wrote down an equality $\sum F(m_1, \ldots, m_m)z^{m_1}\ldots z^{m_m}...
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Galois group of $x^5-5x+10$

I was illustrating the theorem on solvability by radicals through some examples of degree $5$ polynomials. One I chose was $x^5-5x+10$. I was (perhaps wrongly) going to prove that the Galos group is $...
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25 views

4-D lattices and quaternions

It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
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30 views

Sum of divisors in $\mathbb{Z}_2[x]$

Let $A$ denote a polynomial on $\mathbb{Z}_2[x]$ and $\sigma(A)$ denote the sum of divisors on $A$. Let$$A = x^h(x + 1)^k P^l Q^{2n - 1},$$where $P$, $Q$ are irreducible polynomials with degree at ...
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Injectivity of $R \to R[t]/(f)$ for non-constant $f\in R[t]$

Question: Let $R$ be a (unital commutative) ring and $f = a_0 t^n + \cdots + a_n \in R[t]$ a non-constant polynomial. What are (necessary and sufficient) conditions on the coefficients $a_0,\ldots,a_n ...
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46 views

On kernels of commuting operators in infinite dimensions

Let $X$ be an infinite dimensional vector space, and let $\operatorname{S},\operatorname{T}\in\mbox{End}(X)$ be two operators such that: $\operatorname{T}\operatorname{S}=\operatorname{S}\...
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How to recover a $K$-algebra from endomorphism algebra of forgetful functor?

I'm trying to work out how a finite-dimensional $K$-algebra $B$ can be recovered from the algebra of endomorphisms of the forgetful functor $\omega$ from the category of finitely generated left $B$-...
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Shortest proof for showing $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID.

I'm looking for an easy proof for that $\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]$ is a PID. One proof I know is to show that the field norm is a Dedekind-Hasse norm, but this proof is quite dirty( that it ...
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51 views

Maximal ideal containing functions with compact support

I recently proved the following statement: Let $M$ be a smooth manifold and let $I \subseteq C^\infty(M)$ be an ideal such that $C^\infty(M)/I \cong \mathbb{R}$ (such an ideal is clearly maximal, ...
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The Cardinality of a subset of field of 8 elements

Let $F$ be a field of $8$ elements. Let $A$ be a subset of $F$ and $A = \{x \in F \mid x^7=1$ and $x^k \neq 1$ for $k<7\}$. Then find the number of elements in $A$. Argument: Since $F$ is a ...
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72 views

Tropicalization of a line in the projective plane P^2

Lets assume that our field $K$ is the Puiseux series. I have been working with tropicalization from the book "Introduction to tropical geometry" link : https://homepages.warwick.ac.uk/staff/D....
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Is my proof that $ \mathbb{Q}(\sqrt{5}) $ is the only quadratic subfield of $ \mathbb{Q}(\zeta_5) $ correct?

I would like some feedback on my proof of the fact stated in the question. First, we note that $ L = \mathbb{Q}(\zeta_5) $ is the splitting field of $ X^5 - 1 $, so it is Galois. Furthermore, we know ...
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Complex roots of quartic polynomial

This is a question from an undergraduate course on Galois theory: Find all complex numbers which are roots of $P(T)=T^4+2T^2-\sqrt{6}T+\frac{3}{4}$ Can we use Galois theory to solves this? Or ...
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Find the Subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$

As an exercise for myself I want to find the subextensions of $\Bbb Q(\sqrt2, \sqrt3, \sqrt5)/\Bbb Q$. I know that it has Galois group $\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$x$\Bbb Z/2\Bbb Z$, and this has ...
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Completion of Power Series

Let $k$ be field, char. not equal to two. Let $A = k[X,Y]/(Y^2 - X^2(X+1))$ with $\mathfrak{m}=(X,Y)$-adic topology. I want to show that $A'$, the completion of $A$, is isomorphic to $k[[U,V]]/(UV)$....
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50 views

Moments of the number of roots of polynomials over finite fields

Let $F:=\{f\in\mathbb{F}_q[X_1,\ldots,X_n]: \textrm{deg}(f)\leq d\}$ be the set containing all $n$-variate polynomials of degree less than or equal to $d$ over finite field $\mathbb{F}_q$ of prime ...
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39 views

Eisenstein's criterion for two variables

I want to know if there is a criterion, like Eisenstein's criterion, for polynomials with 2 variables? If there is, how does it work?
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“Algebraic” closure of hyperfields

It is very interesting to see that the notion of field (single-valued addition and multiplication) can be extended to the notion of hyperfield (multi-valued addition and single-valued multiplication). ...
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Finite Fields and Discrete Log Problem

The finite field $\mathbb F_{p^p}$, for a prime $p$, can be described as $\mathbb F_p[x]/(x^p−x−1)$. Let $G=\mathbb F_{p^p}^\times/\mathbb F_{p}^\times$. Estimate the cost of obtaining the discrete ...
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What do you call two groups with only trivial homomorphisms between them?

Suppose $G$ and $H$ are groups, and all group homomorphisms $G \to H$ and $H \to G$ are trivial. Is there a common term to describe such a pair of groups with? Like, “$G$ and $H$ are [...]”, or “$G$ ...
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I'm going to do a project on braid groups, and I'm looking for recommendations on books about braid groups.

I asked my advisor (which is not the professor I will be doing the project with) about this project, and he recommended that I read some of Artin's material on braid groups because as my advisor put ...
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Preimage of unit in algebra

If $A,B$ are unital algebras, then for any homomorphism $\psi:A\rightarrow B$ we have $\psi(1_A)=1_B$. What can we say about structure of algebras $A,B$ if the preimage of $\{1_B\}$ is equal $\{1_A\}$ ...
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When modular tensor categories are equivalent?

I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ...
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Abstract algebra: for a polynomial $p$, prove $\sigma(\tau(p))=(\sigma\tau)(p)$ for all $\sigma, \tau \in S_n$

I'm trying to solve to following problem: Part 1: Let $\{x_1,...x_n\}$ be variables. For any polynomial $p$ in $n$ variables and for $\sigma$ $\in S_n$ define $\sigma (p)(x_1,...,x_n)=p(x_{\...
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What is the dimension of the induced module of a linear representation?

Let $V$ be a $n$-dimensional $\mathbb{C}$-vector space and let $G$ be a group. Suppose we have a representation $\phi: G \to \text{Aut}(V)$, then this representation makes $V$ into a $G$-module by $g....
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Proving $End_B(N)$ is Semisimple Algebra

Let $B$ be a semisimple algebra, and $N$ be a finitely generated $B$-module. I am trying to prove $End_B(N)$ is a semisimple algebra. Updated: I changed my attempt based on user26857's and Qiaochu's ...
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Prove that $\phi$ is class preserving automorphism

Let $G$ be a finite group and $\phi:G\to G$ be an automorphism of $G$ such that $\phi|_P=conj(g)|_P$ (restriction of some inner automorphism of $G$) where $P$ is any sylow $p$-subgroup of $G$, then is ...
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Prove that there exist a sylow subgroup of $G$ which is fixed by $\alpha$.

Let $G$ be a finite group and $\alpha\in Aut(G)$ such that $\alpha$ restricted to any sylow subgroup of $G$ equals the restriction of some inner automorphism of $G$ and $(o(\alpha),o(G))=1$, then is ...
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A “new” formula relating the quotients of the upper central series, method of proof and background information

For a finite group $G$ the upper central series is defined inductively by $$ Z_1(G) := Z(G), \qquad Z_{i+1}(G) / Z_i(G) = Z( G / Z_i(G) ). $$ Now I am interested in generalising this formula, i.e. ...
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Characteristic and minimal polynomials

Given $A\in\Bbb Z^{n\times n}$, is it possible to find characteristic and minimal polynomials of $A$ by chinese remainder theorem if we know characteristic and minimal polynomials respectively of $A\...
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Direct sum of $\mathbb{R}(B)$-modules consisting of all cross sections.

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$ let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
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Finite extensions of $\mathbb{Q}((t))$

Let $K$ be a number field. Is every finite extension of $K((t))$ of the form $L((\pi^{1/e}))$, where $L/K$ is finite and $\pi = a_1t + a_2t^2 + \cdots$ for some $a_i\in L$? Is every finite flat ...
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Given a function $\phi$ and the set of iterates $(\phi^t)$ under self-composition, can you construct $\phi^t $ for a real continuous parameter?

$\phi$ is given as any function with the following properties: 1. $\phi$ is strictly increasing 2. $\phi$ is continuous 3. $\phi$ is a mapping of the interval $[0,1]$ into itself. 4. $\phi^n\circ\phi^...
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Is any pair of finite 2-generated perfect groups the quotient of a third finite 2-gen perfect group?

Let $H_1,H_2$ be finite 2-generated perfect groups. Does there exist a finite 2-generated perfect group $G$ which has $H_1,H_2$ as quotients? Of course we may assume $H_1\not\cong H_2$. If $H_1,H_2$ ...
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44 views

Union of conjugates of a subgroup of a finitely generated group.

Is there a finitely generated group $G$, with a proper subgroup $H$, such that $$G=\bigcup_{g\in G}gHg^{-1}$$ I know that this is not the case for a finite $G$: Union of the conjugates of a proper ...
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66 views

Codimensions of $\mathbb{Q}$-subspaces of $\mathbb{R}$

Under the Axiom of Choice, we can pick a Hamel basis $H$ for $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Adjoining all but some elements of $H$ to $\mathbb{Q}$ shows that for any cardinality $1\...
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About centralizers of involutions in finite simple groups

I read that for the classification of the finite simple groups the centralizers of involutions plays a crucial role. This has to to with the Brauer-Fowler-Theorem, which in one formulation could be ...
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Radical of an infinite dimensional Lie Algebra

I am studying Lie Algebras and I just encountered the notion of radical $R$ of a finite dimensional Lie algebra $L$ over a field $F$, the maximal solvable ideal. Since the dimension of $L$ is finite, ...
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Commutability to prove one matrix represented as polynomial of another

For two square matrices $A$ and $B$. If $B$ commutes with all the matrices that commutes with $A$, prove that $B$ can be represented as a polynomial of $A$. It is very complicated if we do it by ...
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60 views

Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
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110 views

Proving $\mathrm{Aut}(S_{n}) = S_{n}$ for $n > 6$

This is a question for a school assignment. We are being asked to prove that $\mathrm{Aut}(S_{n}) = S_{n}$ for $n > 6$. These are the steps we are supposed to follow in our proof. Prove that an ...