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Valuations of Rank
Complete Fields and the Field
Valuation Rings Places
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adic valuation arbitrary archimedian valuation associated valuation ring assume Banach algebra bounded linear functional called canonical expansion Cauchy sequence chapter clearly coefficients complete field complete with respect completes the proof consider cosets denote discrete valuation embedding equation equivalence relation equivalent valuations exists an element exists an integer finite extension field follows immediately group G Hence homomorphism implies integral closure irreducible irreducible polynomial isomorphic leaving k fixed Lemma mapping maximal ideal metric space modulo non-archimedian valuation non-units nontrivial normed linear space null sequence number theory ordered group polynomial positive integer prime quadratic nonresidue quadratic residue quotient ring rank one valuation rational number real number root satisfies sequence with respect subfield subgroup of G subspace suppose Theorem 1.3 trivial valuation unique maximal ideal usual absolute value valuations of Q write