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.

learn more… | top users | synonyms (1)

5
votes
0answers
234 views

System of polynomial equations over rational field

Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations: $u_1v_1+\...
5
votes
0answers
93 views

Profinite group power map

I am working through some exercises in Neukirch's Algebraic number theory and i need some hints for exercise 1 pg 274, it goes as follows: Let $G$ be a profinite group, show that we can extend the ...
5
votes
0answers
80 views

Are there any infinite dimensional subalgebras of the Witt algebra?

The Lie bracket of elements of the Witt algebra is given by: $[L_m,L_n]=(m-n)L_{m+n}$ Are there any infinite dimensional subalgebras of the Witt algebra that are not isomorphic to the Witt algebra ...
5
votes
0answers
103 views

Split short exact sequences and the associated graded algebra

Let $G$ be a group and $K$ be a field. Let $K(G)$ be the group ring and $I$ be its augmentation ideal. You are given that there exists a section of the projection $\pi_s:I^s\to I^s/I^{s+1}$ for all $s$...
5
votes
0answers
221 views

Non-cyclic unit groups

Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor ...
5
votes
0answers
287 views

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: ...
5
votes
0answers
235 views

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. ...
5
votes
0answers
128 views

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 ...
5
votes
0answers
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 ...
5
votes
0answers
213 views

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, $\phi_i$ ...
5
votes
0answers
293 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 ...
5
votes
0answers
368 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 ...
5
votes
0answers
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,...
5
votes
0answers
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 $...
5
votes
0answers
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}...
4
votes
0answers
96 views

Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$?

Is $\pi^\pi$ algebraic over $\mathbb{Q}(\pi)$? I have a feeling that it's a rather easy question, but since my understanding of field extensions is only superficial I really can't handle this ...
4
votes
0answers
56 views

Quote from book

I just read in a book the following dialogue: "'Two negatives make a positive, am I correct?' Andret smiled. 'You are indeed. At least for operations in which the identity element is one.'" Anyone ...
4
votes
0answers
33 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 ...
4
votes
0answers
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 ...
4
votes
0answers
93 views

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 ...
4
votes
0answers
48 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}\...
4
votes
0answers
47 views

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 $B$-...
4
votes
0answers
57 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 ...
4
votes
0answers
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, ...
4
votes
0answers
45 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
4
votes
0answers
37 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 ...
4
votes
0answers
75 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....
4
votes
0answers
30 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 ...
4
votes
0answers
47 views

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 ...
4
votes
0answers
69 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 $k[[U,V]]/(UV)$....
4
votes
0answers
51 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 ...
4
votes
0answers
40 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?
4
votes
0answers
17 views

“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). ...
4
votes
0answers
36 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 ...
4
votes
0answers
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 “$G$ ...
4
votes
0answers
43 views

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 ...
4
votes
0answers
39 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 $\{1_A\}$ ...
4
votes
0answers
47 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 ...
4
votes
0answers
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 (p)(x_1,...,x_n)=p(x_{\...
4
votes
0answers
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 $g....
4
votes
0answers
50 views

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 ...
4
votes
0answers
58 views

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 ...
4
votes
0answers
55 views

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 ...
4
votes
0answers
53 views

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. ...
4
votes
0answers
53 views

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\...
4
votes
0answers
71 views

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 ...
4
votes
0answers
36 views

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^...
4
votes
0answers
47 views

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 $H_1,H_2$ ...
4
votes
0answers
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 ...
4
votes
0answers
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\...