On genus of division algebras

Abstract: The genus

g e n (D)

of a finite-dimensional central division algebra

D

over a field

F

is defined as the collection of classes

[D^{'}] i n B r (F)

, where

D^{'}

is a central division

F

-algebra having the same maximal subfields as

D

. We show that the fact that quaternion division algebras

D

and

D^{'}

have the same maximal subfields does not imply that the matrix algebras

M_{l} (D)

and

M_{l} (D^{'})

have the same maximal subfields for

l > 1

. Moreover, for any odd

n > 1

, we construct a field

L

such that there are two quaternion division

L

-algebras

D

and

D^{'}

and a central division

L

-algebra

C

of degree and exponent

n

such that

g e n (D) = g e n (D^{'})

but

g e n (D o t i m e s C) e g e n (D^{'} o t i m e s C)

.

Cites Work