# 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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 ...
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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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### 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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### 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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### 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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### “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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### 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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### 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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### 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 $H_1,H_2$ ...
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 ...
### 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\...