Brauer group of a field

For a field $F$ , the Brauer group $\operatorname{Br}(F)$ consists of central simple $F$ -algebras modulo the equivalence

A\sim B\quad\Longleftrightarrow\quad M_r(A)\cong M_s(B)\text{ as }F\text{-algebras for some }r,s\ge1.

The group law is $[A][B]=[A\otimes_F B]$ , using the tensor product of algebras. Its identity is $[F]$ , and the inverse of $[A]$ is the class of the opposite algebra $A^{\mathrm{op}}$ .

What the equivalence forgets

The algebras $F$ and $M_n(F)$ have the same Brauer class even though their dimensions differ. More generally, $A$ and $M_n(A)$ represent the same class.

The inverse law comes from

A\otimes_F A^{\mathrm{op}}\cong\operatorname{End}_F(A), \qquad a\otimes b\longmapsto(x\mapsto axb),

whose right side is a matrix algebra.

Quaternion classes

Standard quaternion conjugation gives $B\cong B^{\mathrm{op}}$ . Hence a quaternion class has order at most two: its class is trivial when $B$ splits, and has order two otherwise. This does not mean that two arbitrary quaternion algebras have isomorphic tensor product presentations.

References

John Voight, Quaternion Algebras, author's text, Lemma 8.3.2, Definition 8.3.3, and §8.3.4.