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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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 + ...
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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 ...
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192 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 ...
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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 ...
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44 views

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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36 views

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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59 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 : ...
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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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28 views

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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67 views

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 ...
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49 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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35 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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35 views

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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34 views

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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37 views

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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46 views

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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51 views

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 ...
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56 views

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 ...
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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 ...
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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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$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
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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. ...
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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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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 ...
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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 ...
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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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P.A. Grillet, “Abstract Algebra” linear algebra prerequisites

The author says in the introduction that he only assumes knowledge of linear algebra. I wonder how much linear algebra is actually assumed there? A question for those who have read the book. ...
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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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51 views

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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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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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 ...
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How many $\Bbb{Q}$-algebras of dimension $4$ are there?

How can I construct $\Bbb{Q}$-algebras of dimension $4$. Some examples that comes to mind are $\Bbb{Q} \oplus \Bbb{Q} \oplus \Bbb{Q} \oplus \Bbb{Q}$ , rational quaternions $\Bbb{H}$ and ...
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Checking that a set is a finitely generated ideal

The exercise asks us to prove that $I = \{ f \in \Bbb R[X,Y,Z] \mid f(a,b,c) = 0, ~\forall\,(a,b,c)\in \Bbb S^2 \}$ is a finitely generated ideal of $\Bbb R[X,Y,Z]$. Well, clearly $I$ is an ideal of ...
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Geometric Coset

I am familiar with cosets, but im not sure what to do here (never had to give a "geometric" description of a coset): Consider $\mathbb{R}$ and the subgroup $\mathbb{Z}$. Describe a coset ...
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52 views

Why $ K \cap H $ is a maximal subgroup of $H $?

Suppose $ G $ is a finite group and $ H \unlhd G $. Suppose that $ H \cap M $ is either $ H $ or a maximal subgroup of $ H $ for any maximal subgroup $ M $ of $ G $. Let $ N $ be a minimal normal ...
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A quick Galois Theory question

Let $\mathbb{F}_q$ be a finite field of order $q$ where $q$ is a prime power. For any $d \in \mathbb{N},$ we have an inclusion $\mathbb{F}_{q^d} \subseteq \overline{\mathbb{F}}_q.$ Both ...
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Irreducibility of $p(x)$ implies that of $p(x+c)$ only when taken over a field?

$R$ is a ring and $R[x]$ is the polynomial ring over $R$ . $c$ is any fixed element of $R$ . Then the map $f(x)\mapsto f(x+c)$ is an isomorphism from $R[x]$ to itself. Now ...
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What do we call collections of subsets of a monoid that satisfy these axioms?

Consider a monoid $M$ and a semiring $S$. Then there's an $S$-algebra freely generated by the monoid $M$, which can be described explicitly as the set of all finitely supported functions $M ...
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Diophantine equation: $13^x+3=y^2$

$x,y$ are positive integers. $$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
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35 views

Integer such that there is a $k$-algebra isomorphism for any two algebras.

Is there an integer $\ell = \ell(m, n) \ge 1$ such that for any $k$-algebras $A$ and $B$ there is a $k$-algebra isomorphism $\text{M}_m(A) \otimes_k \text{M}_n(B) \cong \text{M}_\ell(A \otimes_k B)$?
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Finite conjugacy classes in a certain group with three generators

Def. We say that group G has non-trivial finite conjugacy class if there is a conjugacy class $C=\lbrace g_i \rbrace$ such that $g_i \neq 1$ of G with $|C|<\infty$. Let G be group $$<a,b,c ...
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Do we have $\delta(ab)=\delta(a)\delta(b)$ implies $\Delta(cd)=\Delta(c)\Delta(d)$?

Assume that $B$ is an algebra which is also a coalgebras (we do not assume that $B$ is a bialgebra: we do not assume $\Delta(cd)=\Delta(c)\Delta(d)$). Assume that $A$ is $B$-comodule algebra. Then we ...