---
tag: 8A00Y84SW7
kind: counterexample
name: Amitsur's noncrossed product
msc: [16K20, 16S35, 16R20]
uses: [0EA5BTA0CX, 5HRKJBAY2V, NNXSP5KKAC, G21MZQCME9, GJGEVMZ828]
sources:
  - url: https://en.wikipedia.org/wiki/Crossed_product
---

## Counterexample

Let $k = \Q$ and let $n \ge 2$.
Let $X$ and $Y$ be two $n \times n$ matrices whose entries are independent commuting variables over $k$.
The $k$-algebra generated by $X$ and $Y$ is the [[generic matrix algebra|GJGEVMZ828]]; it has no zero divisors.
Its ring of central quotients $\mathrm{UD}(k, n)$ is the *generic division algebra*.
It is a [[division ring|0EA5BTA0CX]] and a [[central simple algebra|5HRKJBAY2V]] of [[degree|NNXSP5KKAC]] $n$ over its center.
Amitsur proved that $\mathrm{UD}(k, n)$ is not a [[crossed product algebra|G21MZQCME9]] when $n$ is divisible by $8$ or by $p^2$ for an odd prime $p$.
That is, for such $n$ it contains no maximal subfield that is a Galois extension of its center.

## Remarks

The idea of the proof is to specialize.
A crossed product structure on $\mathrm{UD}(k, n)$ with group $G$ would pass to every division algebra of degree $n$ over suitable fields of iterated Laurent series.
Amitsur exhibited two such division algebras that are crossed products only for groups of incompatible types, so no single group $G$ works.
For small degrees the opposite holds: every central division algebra of degree $2$, $3$, $4$ or $6$ is a crossed product.
Over local and global fields every central simple algebra is cyclic, so noncrossed products can only appear over more complicated fields.

## Explanations

### L1 human

For a long time it was open whether every finite-dimensional central division algebra comes from the crossed product recipe.
Amitsur found division algebras where this fails.
They are built from two "generic" square matrices of variables, so that they carry no special structure that a crossed product would need.