# 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.

85,672 questions
Filter by
Sorted by
Tagged with
85 views

### Determining If Set is Ring or Field

Decide whether the following subset of $\mathbb{R}$ is not a ring, or is a ring but not a field, or is a field: S = {${a + b\sqrt[3]{2} + c\sqrt[3]{4} | a,b,c \in \mathbb{Q} }$ } I tried approaching ...
11 views

25 views

### Profinite completion of profinite groups

I was trying to prove that $\mathbb{R}/\mathbb{Z}$ cannot be a galoisgroup for any extension. My plan for this was to show that $\mathbb{R}$ is a divisible group and that every quotient of a divisible ...
50 views

26 views

### What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
817 views

### Localization of a module at a prime ideal and local behavior.

Let $M$ be an $R$-module ($R$ commutative with unity) and suppose that given $\mathfrak{p}\subset R$ a prime ideal we have $M_{\mathfrak{p}}\cong R_{\mathfrak{p}}^n$. Is then true that we can find an ...
1 vote
28 views

### $Out(F_n)$ has a free abelian subgroup of rank $2n-3$

Proposition 9.5.4 The group $Out(F_n)$ has a free abelian subgroup of rank $2n-3$. This is a proposition of the book "Topological Methotods in Group Theory" by R. Geoghegan. In the proof he ...
128 views
+50

91 views

### $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
1 vote
50 views

27 views

17 views

### When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$W=\mathbb Z_p[[x,y]]/(xy-c)$$ ...
23 views

### A problem of determining whether a power series belongs to $\mathbb{C}(u)$

I am reading a paper "Drinfeld coproduct, quantum fusion tensor category and applications" and I have a probelm. Here is the arxiv:Drinfeld coproduct, quantum fusion tensor category and ...
57 views

### Simple module is isomorphic to $R/M$ where $M$ is a maximal ideal
Let $E: Y^2=X^3+Ax+B$ be an elliptic curve, defined over $\mathbb{F}_p$ where $p$ is a prime. Define: $$\phi: E(\bar{\mathbb{F}}) \rightarrow E(\bar{\mathbb{F}})$$ by \phi(P) = \left\{ \begin{array}...