# Questions tagged [abstract-algebra]

For questions about monoids, groups, rings, modules, fields, vector spaces, algebras over fields, various types of lattices, and other such algebraic objects. Associate with related tags like [group-theory], [ring-theory], [modules], etc. as necessary to clarify which topic of abstract algebra is most related to your question and help other users when searching.

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### Where Does This Proof Use $\text{Char}(K)=p$?

$\DeclareMathOperator{\char}{char}$ $\DeclareMathOperator{\aut}{Aut}$ I've been reading over the following theorem (5.7.7) from Hungerford's algebra, and two things are confusing me about the proof of ...
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### use of binary operation to represent hadamard transform

In very frank terms, meaning basic binary operations and well-defined definitions, can someone explain to me what this operation is supposed to mean and what the notation is supposed to represent? I ...
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### How can you have an isometry without a metric/coordinates?

I'm self-studying Artin's Algebra and I'm getting a bit thrown through a loop on isometries (in Chapter 6, in case you're curious). Specifically, a section about change of coordinates: Let $P$ denote ...
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### Proving the Relative Nullstellensatz

Let $X \subseteq \mathbb{A}^n$ be an affine subvariety and let $A(X)$ be the corresponding coordinate ring. For $Y \subseteq X$ and $S \subseteq A(X)$ we define the following relative analogues to the ...
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### Show that the image of an irreducible affine variety is irreducible [duplicate]

I would really appreciate some help on this: Definition: Let $k$ be a field. An affine variety is a space with functions $(X, O_X)$ i.e. a topological space equipped with a sheaf of $k-algebras$, ...
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### Definition of Presentation of Group in Dummit’s Abstract Algebra [closed]

A subset $S$ of elements of a group $G$ with the property that every element of $G$ can be written as a (finite) product of elements of $S$ and their inverses is called a set of generators of $G$. We ...
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### Characteristic of a quotient ring in $\mathbb Z[i]$ [duplicate]

I would like to solve this exercise: Let $\mathbb Z[i]$ be the Gaussian integers ring. Check whether the ideal $I$ generated by $2−i$ is prime in $\mathbb Z[i]$. Find the characteristic of quotient ...
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### How to write Type III matrix as a product of Type I matrices?

I saw this question Prove that, over a Euclidean domain $R, I + cE_{ij}$ generate $SL_n(R)$. and in the comment section, someone mentioned that you can do something about type III matrices. I tried to ...
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### Zero divisors in $\mathbb Q [T]/(T^4+T^2+1)$

I am asked to find all zero divisors in the quotient ring $\mathbb Q [T]/(T^4+T^2+1)$. The most that I have come to is to take into account that if that set is a field and the polynomial is ...
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### If Galois group is $D_8$ then show that an intermediate extension has quadratic subfield

Given that $F$ is degree extension $4$ of $\mathbb Q$ which is not Galois. If $K$ is Galois closure of $F$ and $\operatorname{Gal}(K/\mathbb Q)$ is $D_8$ then I have to show that $F$ has subfield that ...
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### Algebraic structures with opposite subalgebra lattices and congruence lattices that are not groups with operators

Are there any algebraic structures where the congruence lattice and subalgebra lattice are opposites that do not look like groups with operators? My motivation for this question is noticing that ...
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### Noetherian rings and greatest common divisors of ideals.

I'm reading this Algebra book at the moment and found an exercise that I've been struggling with for quite a while now: Let $R$ be a ring and $M$ be the smallest set containing all principal ideals of ...
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### Describe an ideal in a residual polynomial ring

Given the polynomial ring over a field $\mathbb{K}[x]$ and $f\in \mathbb{K}[x],$ one may define the residual polynomial ring $\mathbb{K}[x]/(f).$ I am wondering how does an ideal of the residual ring ...
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### Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent?

Which general methods are there for determining if $t$ and $e^t$, where $t$ is any given transcendental number, are algebraically independent? Can Schanuel's conjecture be used for this? We have ...
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### Let $F$ be a field. Every (non-constant) polynomial $p \in F[X]$ is transcendental over $F$

I will start with the Definition: Let $L/K$ be a Field extension. The element $a \in L$ is called algebraic over $K$ if there exists a polynomial $p \in K[X]$ such that $p \neq 0$ and $p(a)=0$. If ...
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### If $B$ is an integrally closed domain and $B \to A$ is an integral extension, then $A$ is the integral closure of $B$ in $K(A)$ or not?
Here $K(A)$ and $K(B)$ mean the fraction fields of $A$ and $B$, respectively. I see that the integral closure of $B$ in $K(A)$, denoting it as $C$, satisfies $A \subseteq C$ for $B\to A$ being an ...