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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

8 votes
0 answers
311 views

Passing to torsion of an exact sequence

If $$ \Theta\colon\quad 0\to A\to B\to C\to 0 $$ is an exact sequence of abelian groups, and $n$ is an integer, then one obtains an exact sequence $$ 0\to A[n] \to B[n] \to C[n] \stackrel{\delta_n(\Th …
Alex B.'s user avatar
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5 votes

Examples of DVRs of residue char p and ramification e

Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, …
Alex B.'s user avatar
  • 13k
1 vote

Conjugacy for p-adic matrices of finite order II

Let me try this. I think that the answer this time is positive. Step 1: We will first reduce to $p$ power order. Let $M$, $M'$ be matrices over $\mathbb{F}_p$ of order $p^na$, where $p\nmid a$. Then, …
Alex B.'s user avatar
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13 votes
Accepted

Conjugacy for $p$-adic matrices of finite order

I think I finally have a correct answer for arbitrary $p$. As F. Ladisch notes, $G=C_{p^3}$ has only finitely many indecomposable modular representations. For the following argument, I will not only n …
Alex B.'s user avatar
  • 13k
4 votes

Is it possible to recover the degree of a field extension from a list of elements and the gr...

To put this one to rest, I will answer the more precise question that, after much prodding, we got Adam to formulate in the comments. I am merely paraphrasing a comment of Qiaochu. If you are given t …
Alex B.'s user avatar
  • 13k
14 votes

Algebra with a certain abelian group as the multiplicative group

I am going to assume that by "algebra" you simply mean a ring. The answer is "no", in general. For example $\mathbb{Z}/5\mathbb{Z}$ is not the unit group of a ring. Indeed, suppose it was the unit gro …
Alex B.'s user avatar
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