By Professor André Weil (auth.)

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The appropriate concept is here as follows: DEFINITION 3. By an R-lattice in a vector-space V of finite dimension over an R-field, we understand a discrete subgroup L of V such that VIL is compact. We have to recall here some elementary facts about discrete subgroups. Let G be a topological group, r a discrete subgroup of G, and

This is clear for n = 1. For n> 1, use induction on n. By corollary 3 of prop. 1, we may choose V1 so that the space V1 generated by V1 is Northogonal to W z ; then, by prop. 2, N(v~ +w z )=suP(N(V'1),N(wz)) whenever V'1 E V1, W z EJrJ-l. Applying the induction assumption to the K-norm induced by N on Wz , and to the sequence Wz , ... , w", we get our result. COROLLARY. To every subspace W of V, there is a subspace W' which is N -orthogonal to W. Take a sequence W1 , ••• , w", as in proposition 3, such that W is one of the spaces in that sequence, say ~.

Take notations as in theorem 7; corollary 1 of tho 7 shows that M is a field with q elements. Taking ~ = 1t and A = M in corollary 2 of tho 6, we get for every XEK with ord(x)~n a unique series expansion +00 _'" xL. J-l i 1t,i i=n with J-liE M for all i ~ n. One verifies at once that the rules for the addition and multiplication of such series are the usual ones for formal powerseries in algebra (or for convergent power-series in classical analysis). Moreover, this is an isomorphism also in the topological sense if the field of formal power-series is provided with its usual topology, that for which the ring Ro of "integral" power-series (those containing no power of the indeterminate with an exponent < 0), and the ideals generated in it by the powers of the indeterminate, make up a fundamental system of neighborhoods of O.

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