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Where (**) usual (finite) Example 8. if a is a positive expansion p-adic (*) a = j ~ Addition of a fields. field of p-adic numbers anP n Example ~p, which ~Qp pletion p. consists Q interested of all formal are d~fined with some care. p ~an pn in the expressions and "carry For example, and ~bn pn it may be necessary I" (compare the last remark in The reader may supply a more careful is a field of will be the (0 < a n < p, a N ~ 0). an+b n modulo 7). (*) We will be particularly and multiplication to reduce by: integer, to the base in order to add two expressions that (0 <_ a i < p) the coefficients a i are best described n a-i_~ipl has order at least n+1.

N ~ E a ~I F O. Here we have use~ the fact that ~ aj = 0 F for j < q-l, an elementary fact which is easily verified. Theorem 39. ,Xn~ mial over in K of degree d trivial zero in Proof: n > d2 F(t) where F be a homogeneous polyno- variables. Then p has a non- K n. We seek a solution of the form xi = Yio + Yil t + "'" + Yis ts with ~ij ~ F. We will eventually choose s clear the denominators from the coefficients that p has coefficients rather large. of p, in the polynomial ring the maximum of the degrees of the coefficients assuming therefore Fit].

A2~p for all 2. P 3. -q~P. 4. -I ~ P. 5. For any is closed u n d e r +,-. a The c o n s t r u c t i o n Let aEF. Po in of F either P be the smallest a proceeds subset or -a is in P. as follows: of F satisfying I-3. Using the 30 fact that Extend q Pc in F, one computes by Zorn's Lemma to a maximal and verify w is not a sum of squares subset of F that -I ~ P o . satisfying I-4, that 5 then holds. Notes. Lemma 2 is the poor man's Lefschetz version of the full Lefschetz of Lemma 2 to prove it is noted morphisms Theorem of algebraic ~, in passing where a The application in [5 ~, where equally well to injective endo- varieties.

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