The Theory of Valuations

abelian extension according additive apply assert assume basis belonging called Chapter characteristic coefficients commutative complete complete field Consequently consider consists construction contains Conversely COROLLARY corresponding cyclic cyclic extension define definition denote determined discrete division algebra elements equal equation exists extension of F factor field F Finally finite first follows function Galois group given Hence holds homomorphism hypothesis implies inertial infinite integers isomorphic least Lemma lies limit Math maximal maximal order Moreover multiplicative norm normal normal extension observe obtain p-adic polynomial positive preceding prime ideal prolongation Proof properties prove pseudo rational relation relatively Remark representatives residue class field respect roots of unity separable sequence simple subfield subgroup suitable Suppose Theorem unique unit valuation ring value group