5
$\begingroup$

Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a matrix ring over $D$?

(Note that the answer with general base field is no, as there are countably many finite groups and each group algebra has only finitely factors in the Wedderburn decomposition, so the division algebras which appear as Wedderburn factors are countable, thus any field with uncountably many finite extensions is a counterexample)

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ The results of the paper ac.els-cdn.com/0022314X73900437/… would seem to indicate that not all appear. $\endgroup$
    – Benjamin Steinberg
    Commented May 31, 2018 at 2:20
  • $\begingroup$ @BenjaminSteinberg this answers the question, thanks! $\endgroup$
    – Lukas Heger
    Commented May 31, 2018 at 2:23
  • 2
    $\begingroup$ A standard reference for this topic is Yamada, The Schur subgroup of the Brauer group, LNM 397, Springer (1974), doi 10.1007/BFb0061703. $\endgroup$
    – Frieder Ladisch
    Commented May 31, 2018 at 12:56

1 Answer 1

Reset to default
10
$\begingroup$

The subgroup of the Brauer group generated by (and in fact, consisting of) such division algebras is called "the Schur group". Brauer-Witt theorem asserts that it is given by cyclotomic algebras, so the answer is negative. I learned this here.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .