# 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 is predicated on a path taught at Princeton college in 1961-62; at the moment, an outstanding set of notes was once ready through David Cantor, and it was once initially my purpose to make those notes on hand to the mathematical public with in simple terms fairly minor adjustments. Then, between a few previous papers of mine, I by chance came upon a long-forgotten manuscript by way of Chevalley, of pre-war classic (forgotten, that's to claim, either by means of me and via its writer) which, to my flavor no less than, appeared to have elderly rather well. It contained a short yet primarily com plete account of the most positive factors of classfield thought, either neighborhood and international; and it quickly turned seen that the usefulness of the meant quantity will be tremendously more suitable if I integrated one of these remedy of this subject. It needed to be increased, based on my very own plans, yet its define might be preserved with out a lot swap. actually, i've got adhered to it relatively heavily at a few serious points.

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**Example text**

7 to ~ and Ln~, and also to ~ and L'n~; this gives Vi such that Ln~=Rvi and v; such that L'n ~=Rv;. Writing v;=xjVj with xjeKx, and putting vj=ord(xj), we get integers Vj with the required property. N'(v)~1. COROLLARY 1. Let V and L be as in theorems 1 and 2, and let M be an R-module in V. Then there is a basis {vt> ... ,vn } of V over K, and there are integers r, sand Vt,. •. ,V" such that O~r~s~n, L=IRvj and , s L: M= L:rJvj + Kv". j=t ,,=,+t Let W be the subspace of V generated by M, and W' the maximal subspace contained in M; call s the dimension of W, and r the codimension of W' in W.

11:" - l} is a basis of K as a left vector-space over K 1 , that it generates R as a left Rcmodule, and that nd is a prime element of K 1, hence also of Ko since it lies in Ko. As this implies that the order of ramification of Kover Ko is d, it completes the proof. Notations being as in proposition 5, let qJ be the Frobenius automorphism of K lover Ko; as this also generates the Galois group of K 1 over K o, we must have qJ = rx,r, with r prime to d and uniquely determined modulo d. It will be shown in Chapter XII that, when Ko is given, d and r may be chosen arbitrarily, subject to these conditions, and characterize the structure of the division algebra K uniquely; in other words, two division algebras of finite dimension over K o, with the center K o, are isomorphic if and only if they have the same dimension d2 over K o, and the integer r has the same value modulo d for both.

Moreover, if Wl = V, W2 , ... , w,. is a sequence of subspaces of V such that W; is a subspace of W; _ 1 of codimension 1 for 2 ~ i ~ n, then the V; may be so chosen that W; = V; + ... + v,. for all i. This is clear for n= 1. For n> 1, use induction on n. By corollary 3 of prop. 1, we may choose Vl so that the space Vl generated by V l is Northogonal to W2 ; then, by prop. 2, N(V'l +w2)=sup(N(v~),N(W2)) whenever V'l e Vl , W2 e ~. Applying the induction assumption to the K -norm induced by N on W2 , and to the sequence W2 , ...