Math 250a: Higher Algebra
Problem Set #9 (6 December 2004):
Quaternion algebras
A bit more about Baer multiplication:
1. Let Abe an abelian group and Gany group acting on A. For any extension
1→Aι
→Eπ
→G→1 consistent with this action, let Eobe the extension
1→A−ι
−→ Eπ
→G→1 with the opposite embedding of Ain E. [Why do
E, E ohave the same G-action on A?] Prove that Eois the inverse of Ein
two ways: by identifying (E, E o)/Q with the semidirect product AoG,
and by showing that E, Eocorrespond to inverse elements of H2(G, A).
Note that the formula for Eois what one might expect from the special case
(G, A) = (Gal(L/k), L∗) and our results about the opposite of a central simple
algebra.
In the next two problems, we describe generalized quaternion algebras over an
arbitrary field knot of characteristic 2.
2. i) For any (commutative) field k, define a map x7→ ¯xon M2(k) by ¯x=
Tr(x)·1−x. Here Tr(x) is the trace of xas a 2 ×2 matrix, and 1is
the 2 ×2 identity matrix, which is the unit element of M2(k). Prove that
this map is an anti-involution, i.e., that it satisfies the identities ¯
¯x=x
and xy = ¯y¯x. [This can be done either by explicit computation or via a
relation between ¯xand the transpose of x.]
ii) Now suppose that A/k is any central simple algebra with dimkA= 4.
Define a map x7→ ¯xon Aby ¯x= Tr(x)·1−x, where Tr(x) is the reduced
trace of xand 1 is the unit element of A. Prove that this map is an
anti-involution.
Let A0be the kernel of Tr; it is a k-vector subspace of Aof dimension 4 −1 = 3.
Let N:A→kbe the reduced norm, so N(x) = x¯x. This is a quadratic form
on A, and the associated bilinear form is
(x, y) = N(x+y)−N(x)−N(y) = x¯y+y¯x= Tr(x¯y).
Note that if x∈A0then N(x) = −x2.
3. i) Prove that if x, y ∈A0with N(x) = N(y)6= 0 then x, y are conjugate
in A.
ii) Prove that there exist i∈A0with N(i) nonzero. Fix one such i, and
let c=N(i). Since also N(−i) = N(i), by part (i) there exist invertible
z∈Asuch that iz=−zi. Show that i¯z=−¯zi, and hence that ij =−ji
where j:= z−¯z. Show that j∈A0and j6= 0.
iii) Now let k=ij =−ji. Show that k∈A0and ki =−ik =cj. Let d=
N(j) = −j2, and determine jk,kj,k2in terms of c, d, i,j,k. In particular
show that i,j,kare pairwise orthogonal for the bilinear form (·,·).
iv) If Ais a division algebra, prove that i,j,kare linearly independent, and
