## Basic Number Theory by Professor André Weil (auth.)

By Professor André Weil (auth.)

)tPI(}jlOV, e~oxov (10CPUljlr1.'CWV Aiux., llpop. . .dsup.. the 1st a part of this quantity relies on a path taught at Princeton college in 1961-62; at the moment, an outstanding set of notes used to be ready through David Cantor, and it used to be initially my purpose to make those notes to be had to the mathematical public with in simple terms fairly minor adjustments. Then, between a few previous papers of mine, I by accident stumbled on a long-forgotten manuscript through Chevalley, of pre-war classic (forgotten, that's to claim, either via me and by means of its writer) which, to my flavor at the very least, looked as if it would have elderly rather well. It contained a short yet basically com plete account of the most positive aspects of classfield concept, either neighborhood and international; and it quickly grew to become visible that the usefulness of the meant quantity will be tremendously superior if I incorporated any such therapy of this subject. It needed to be accelerated, in keeping with my very own plans, yet its define may be preserved with out a lot swap. in truth, i've got adhered to it really heavily at a few serious points.

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1 of§ 1, if N Mis a K-norm, M is compact. Conversely, assume that M is compact, and take v-=foO; then Mv is the subset of K corresponding to (Kv)nM under the isomorphism x ~ x v of K onto K v; therefore M v is compact and cannot be K. This completes our proof. COROLLARY 1. An open R-module M in V is a K-Iattice if it contains no subspace of V other than O. e. KvcM. Conversely, as every subspace of V, other than 0, is closed in V and not compact, no such subspace can be contained in M if M is compact.

8. PROPOSITION 10. Let K be a commutative p-field of characteristic p. Then 1 + P, as a Zp-module, is the direct product of a countably i1ifinite family of modules isomorphic to Zp. 35 Lattices over R § 4. By tho 8 of Chap. 1-4, we may regard K as the field of formal powerseries in one indeterminate 1t, with coefficients in the field F q with q = pI elements. Here it is easy to give explicitly a family of free generators for the Zp-module 1 +P. In fact, take a basis {1X 1 , ... ,IX/ } for Fq over the prime field F p" As generators of 1 + P, we take the elements 1 + IX/X", where 1 ~ i ~ f, n running through the set of all integers > 0, prime to p.

This follows at once from the inequality N(x' v' -xv) ~ sup(modK(x')N(v' - v), modK(x' - x)N(v») which is an immediate consequence of def. 1. Therefore N is continuous, and the sets Lr make up a fundamental system of closed neighborhoods Norms §l. 25 of 0; in particular, Lr must be compact for some r>O. Now, for any s>O, take aEK x such that modK(a) ~ r/s; then, as one sees at once, L. is contained in a-I Lr; therefore it is compact. COROLLARY 1. There is a compact subset A of V - {O} which contains some scalar multiple of every v in V - {O}.