Division Algebras over Local Fields
This chapter gives a fairly complete description of the finite dimensional division algebras over fields that are locally compact in the topology of a discrete valuation, that is, local fields. The most important property of these algebras is that they contain maximal subfields that are unramified extensions of their centers. It follows that all such algebras are cyclic. Moreover, the classification of the unramified extensions of local fields gives a characterization of the Brauer groups of such fields; they are all isomorphic to ℚ/ℤ.
KeywordsLocal Field Division Algebra Valuation Ring Galois Extension Principal Ideal Domain
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