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G such that the image j(G) is dense in G, j is a homeomorphism G -> j (G), any continuous homomorphism f : G -> G' into a complete group G' can be uniquely factorized as f = g o j : G --p G - G' with a continuous homomorphism g : G -> G'.

4. 3) immediately shows that the projective limit (E, 4'n) is a topological space equipped with continuous maps >U : E -> En having the universal property with respect to continuous maps. Any topological space X equipped with a family of continuous maps fn : X -> E such that f, = (pn o fn+1 (n > 0) has the factorization property fn = 4'n o f with a continuous function f : X E. Indeed, this factorization ... is simply given in components by the f,, and is continuous by definition of the product topology (and the induced topology on the subset lim En C fl E,,).

1. First Principle Let us explain the first principle in a particular case. Let P(X, Y) E Z[X, Y] be a polynomial with integral coefficients. When speaking of solutions of the implicit equation P = 0 in a ring A, we mean a pair (x, y) E A x A = A2 such that P(x, y) = 0. Proposition, The following properties are equivalent: (1) P = 0 admits a solution in Zp. (ii) For each n > 0, P = 0 admits a solution in Z/ p" Z. (iii) For each n > 0, there are integers a", b" such that P(an,b,,)=- 0modp". PROOF.

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A Course in p-adic Analysis (Graduate Texts in Mathematics) by Alain M. Robert

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