. Let F be a field and define the ring F((x)) of formal Laurent series with coefficients from F by F((x)) =ax | an F and N Z). (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 N 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).