0
votes
0answers
67 views

What is free product of two k-algebra (k is a field) [closed]

Let A, B be k-algebra (k is a field). What is free product of k-algebra A and k-algebra B? ($A \ast_{k}B$)
1
vote
2answers
294 views

What structure supports division to a unique quotient and remainder?

This has been bugging me for a while. According to https://en.wikipedia.org/wiki/Euclidean_division, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r ...
2
votes
1answer
203 views

(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)

Let $K$ be a skew-field, infinite dimensional over its center $F$. From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive ...
11
votes
2answers
669 views

units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to me when writing ...
13
votes
4answers
940 views

Dimension of central simple algebra over a global field “built using class field theory”.

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following: $$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$ The ...