# Tagged Questions

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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### 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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### 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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### 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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### Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
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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 ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### How to apply a double centralizer property on a faithful module of a self-injective Artin algebra?

Let all considered algebras be Artin algebras and let all considered modules be finitely generated. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: ...
### For a $3$-by-$3$ matrix, find $A$, $B$ integer $3$-by-$3$ matrices with determinant $\pm 1$ such that $AMB$ is diagonal.
For a $3$-by-$3$ integer matrix $M$, find $A$, $B$ integer $3$-by-$3$ matrices with determinant $\pm 1$ such that $AMB$ is diagonal. $\textbf{My Attempt}:$ Given $M$ we want to right-mulitply by an ...
### Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?
Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$. This is very easy to prove if the ring is unital as you may write ...