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Additional info for 250 problems in elementary number theory

Example text

8) is a metric of the field k. 7. 8) is isomorphic to the field k {t} of formal power series, which consists of all series of the form (an E k o ) under the usual operations on power series (the integer m may be positive, negative, or zero). 5. 1. Congruences and Equations in the Ring Op At the beginning of Section 3 we considered the question of the solvability of the congruence x 2 == 2 (mod 7") for n = 1,2, ... , and this led us to the concept of a p-adic integer. 1). This connection is described more fully in the following theorem.

If F(x 1 , ... ,xn ) is an absolutely irreducible polynomial with rational integer coefficients, then the equation F(x 1 , ... , x n ) = 0 is solvable in the ring 0 p of p-adic integers for all prime numbers p greater than some bound which depends only on the polynomial F. 5) is solvable for all k. 5) for all p to the question of the solvability of the equation F = 0 in the ring Op for a finite number of primes p. We shall not deal here with the question of the solvability of the equation F = 0 in the ring Op for these finitely many p (for the case of quadratic polynomials this will be done in Section 6).

The equation F(x 1, ... , x n) = 0 has a nontrivial solution in the ring Op if and only if for every m the congruence F(Xll ... , x n) == 0 (mod pm) has a solution in which not all terms are divisible by p. It is clear that in Theorems 1 and 2 Fmay be a polynomial whose coefficients are p-adic integers. 2. 1). It is generally difficult to tell when we may limit our consideration to only a finite number of these. Here we shall consider a special case. Theorem 3. Let F(x 1, ... , x n) be a polynomial whose coefficients are padic integers.