By Waclaw Sierpinski, I. N. Sneddon, M. Stark
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Additional info for A Selection of Problems in the Theory of Numbers
A,x-“; an < 0. § 4. Structure here all the terms in the right-hand of p-fields side are in the maximal 23 ideal P of R, so that 1 EP, which is absurd. Conversely, let x be any element of R. By corollary 2 of th. 3,s 2, K has a finite dimension over K,; therefore, if we put K’ = K,(x), this is a commutative field and a finite extension of K,. Call F the irreducible manic polynomial, with coefficients in K,, such that F(x) = 0; in some algebraic closure of K’, call K” the field generated over K, by all the roots of F, so that F splits into linear factors in K”.
If L is any closed subgroup of x the subgroup L, of V+ associated with L by duality consists of the elements u* of I” such that (u,u*)~= 1 for all UEL; in view of (8), this implies that, if L is a left module for some subring of K, L, is a right module for the same subring, and conversely. In particular, if K is a p-field and R is the maximal compact subring of K, L is a left R-module if and only if L, is a right R-module. As we have seen that L is compact and open in I/’ if and only if L, is so in V*, we see that L is a K-lattice if and only if L, is one.
If K is commutative, proposition 6 may be expressed by saying that R is the integral closure of R, in K. Chapter II Lattices and duality over local fields 0 1. Norms. In this 9 and the next one, K will be a p-field, commutative or not. We shall mostly discuss only left vector-spaces over K; everything will apply in an obvious way to right vector-spaces. Only vector-spaces of finite dimension will occur; it is understood that these are always provided with their “natural topology” according to corollary 1 of th.
A Selection of Problems in the Theory of Numbers by Waclaw Sierpinski, I. N. Sneddon, M. Stark