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Example text

If D = k we are done. This leaves the unique quaternion division algebra D to be considered. The case that the involution θ on the quaternion division algebra D is of first kind is easy, since we already know that D has a main involution, so by Skolem-Noether any other involution of first kind differs by a conjugation. Now suppose that θ is of second kind. Let \alf → α be the main involution. Then α → (αθ ) is an automorphism of order 2 of D and gives a non-trivial automorphism τ on k. The set Do = {x ∈ D : xθ } is a subring of D containing the subfield ko of k fixed by τ .

1 are linearly independent over k. Thus, k(π) is a subfield of D, with [k(π) : k] = ν. Since any subfield of D has degree over k less than or equal d, we find ν ≤ d. For the same reason, [K : k] ≤ d. At the same time, the existence of expressions i αi π i for elements of D shows that nν ≥ d2 . Thus, n=ν=d That is, K is indeed a maximal subfield of D. /// Corollary: Over a non-archimedean local field k there is a unique quaternion division algebra up to isomorphism. Proof: We know that any quaternion division algebra is obtained as a cyclic algebra over the unique unramified quadratic extension K of k, with cocycle f (σ i , σ j ) = 1 36 for i + j < 2 for i + j ≥ 2 Paul Garrett: Algebras and Involutions (February 19, 2005) where is a local parameter in k.

Now we return to the proof of the theorem. Let τ be the restriction to k of the involution θ. As usual, the existence of the involution gives an isomorphism Dopp ≈ Dτ of k-algebras, where now (unlike an earlier discussion) we allow for the possibility that θ is not trivial on the center k. By the proposition, D ≈ Dτ as k-algebras, so we conclude that D ≈ Dopp as k-algebras. By the structure of the Brauer group, this implies that the similariy class of D in the Brauer group Br(k) is of order a divisor of 2.

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