# Questions tagged [abstract-algebra]

For questions about groups, rings, fields, vector spaces, modules and other algebraic objects. Associate with related tags like group-theory, ring-theory, modules, etc. to clarify which topic of abstract algebra is most related to your question and help other users when searching.

10,357 questions
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### If polynomials are almost surjective over a field, is the field algebraically closed?

Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. ...
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### What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for ...
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### Where Fermat's last theorem fails

It's fairly well known that Fermat's last theorem fails in $\mathbb{Z}/p\mathbb{Z}$. Schur discovered this while he was trying to prove the conjecture on $\mathbb{N}$, and the proof is an application ...
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### Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we should ...
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### Why $E\otimes_KE\cong EG$ implies that Galois theory works?

This is a part of statement in the book I do not fully appreciate. Suppose $E/K$ is Galois extension and $G$ the galois group of $E/K$. $E[G]$ is the group ring formed by finite group $G$. "It is ...
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### Galois correspondence and characteristic subgroups

It is well-known that Galois correspondence sends a normal subgroup to a normal extension of a field. Specifically, given a Galois extension $L/K$ and the corresponding Galois group $G$, normal ...
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### Proving a ring in which $r^n=r$ for all $r$ is commutative.

Let $R$ be a ring with identity such that there is a positive integer $n\geq 2$ for which $r^n=r$ for all $r\in R$. Prove $R$ is commutative. I had proven before that If $n=2$ it is commutative as ...
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### On a Proof in Galois Theory

We have the following lemma (used in the proof of Abel's Theorem): If $\text{Char}(F)=0$, $E/F$ is a radical extension, and $K/E$ is the Galois closure of $E/F$, then $K/F$ is also a radical ...
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### Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
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### Method of showing maximal number of intermediate field extensions of a Galois extension with given degree

The task is the following: Show that a Galois extension $L/K$ of degree $45$ has got at most $12$ intermediate field extensions. Below I present a proof. I seek a more general method for this ...
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### Finding a chain map from an $F$-acyclic resolution to an injective resolution which is a monomorphism in each degree

Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F:\mathcal{A}\to\mathcal{B}$ be a left-exact additive functor to $\mathcal{B}$ another abelian category If $M$ has an $F$-...
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### How to effectively compute fundamental units in rings?

Consider the ring of integers of $K=\mathbb{Q}(\sqrt 2,\sqrt 3)$. By Dirichlet's unit theorem the units of $\mathcal O_K$ have rank 3, so they are expressible as $\pm u_1^au_2^bu_3^c$ for suitable ...
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### A Ramanujan-type trigonometric identity

At the end of the following article: http://www.ijpam.eu/contents/2013-85-1/15/15.pdf It is asserted that the russian mathematician, Sergey Markelov, in private communication, told them that he ...
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### Every polynomial's image contains $0$ or $1$ in a field $\Bbb F$

This question talks about fields in which every polynomials are almost surjective, while I am interested in the following case: $\Bbb F$ is a field such that for every non-constant polynomial $f$ ...
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### Prove that the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated

Knowing that for $n \geq 2$, $\mathrm{GL}(n, \mathbb{Z}) = \big\{ A \in \mathrm{M}_{n,n}(\mathbb{Z}) \mid \det(A) \in \{ 1, −1 \} \big\}$ is a group with respect to matrix multiplication, prove that ...
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### In which algebraic theories do 'free' and 'projective' coincide?

Free models of algebraic theories are always projective objects in the category of models, but the converse is not always true. For instance, some (actually, all) projective modules are direct ...
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### Classification of all subrings

Let $R$ be an integral domain whose underlying additive group is finitely generated free and whose field of fractions $K$ is a finite Galois extension of $\mathbb{Q}$. Is there a method of ...
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### An equivalence between group cohomology and sheaf cohomology

I'm recently reading group cohomology from Serre's book local fields, and he uses there the following terminology $H^q(G,A)$ the q-th degree cohomology of G with coefficent in $A$. So i started to ...
### Subgroups of $GL_n$ and group actions
For my abstract algebra class I have to do some exercises concerning group actions of $GL_n(K)$ on the set of flags F_n = \{0 \subseteq V_1 \subseteq V_2 \subseteq \ldots \subseteq V_r = K^n \,\vert\...