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By Garrett P.

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Example text

Now we can use the general algebraic results on cyclic algebras to give a complete description of the Brauer group of a non-archimedean local field. We recall that the cyclic Galois group of an unramified extension K of a non-archimedean local field k, where the latter’s residue class field has q elements, is generated by the Frobenius automorphism σ defined by ασ = αq mod m where m is the maximal ideal in the integers o of k. Theorem: The Brauer group Br(k) of a non-archimedean local field k is canonically isomorphic to Q/Z.

That is, from the structure theory of division algebras over local fields, D is of dimension 1 = 12 or 4 = 22 . 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.

For there to exist β ∈ K so that pi = NormK/k β it is necessary (though not sufficient) that there be a prime ideal in K lying over p with residue class field extension of degree i over Z/p, but we have arranged that the residue class field extension degree is n. Thus, pi is not a norm for 1 ≤ i < n, and A(p) is a division algebra. Thus, by the proposition, A(p) has involution of second kind with central fixed field ko . Remark: The above construction of non-trivial division algebras with involutions of second kind fails over local fields such as Qp , since for p = 1 mod n the nth roots of unity already lie inside Qp , so we do not obtain a dihedral Galois extension in the first place.

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