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Let F be a field and define the ring F x of formal Laurent series...

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Algebra

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Let F be a field and define the ring F x of formal Laurent series with coefficients from F by F x anx an F and N Z n2N Every element of F x is a power series in x plus a polynomial in 1 x i e each element of F x has only a finite number of terms with negative powers of x a Prove that F x is a field b Define the map v F x Z by v anx N 2N where an is the first nonzero coefficient of the series i e N is the order of zero or pole of the series at 0 Prove that v is a discrete valuation on F x whose discrete valuation ring is F x the ring of formal power series cf Exercise 26 Section 1

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