By Paula Ribenboim

ISBN-10: 0471718041

ISBN-13: 9780471718048

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**Additional resources for Algebraic Numbers (Pure & Applied Mathematics Monograph)**

**Example text**

Squares in the Field of p- Adic Numbers For the study of quadratic forms over a given field it is important to know which of the elements of the field are squares. Therefore we first turn to the 48 CONGRUENCES [Chap. 1 study of squares in the field R, of p-adic numbers. We know (Section 3, Theorem 4) that every nonzero p-adic number a can be represented uniquely in the form a = pm&,where E is a p-adic unit (that is, E is a unit in the ring 0, of p-adic integers). If a is the square of the p-adic number y = p k c o , then m = 2k and E = eO2.

Alp, a. + alp + a2p2,... 1, where 0 < a , < p . On the other hand, every sequence of this type is a canonical sequence, which determines somep-adic integer. From this it follows that the set of all canonical sequences, and also the set of all p-adic integers, have the cardinality of the continuum. 2. The Ring of p-Adic Integers Definition. Let thep-adic integers cc and p be determined by the sequences {x,} and {y,}. Then the sum (respectively, product) of o! and B is the p-adic integer determined by the sequence {x, y,,} (respectively, {x,~,,}).

Xnp1,a) is not identically congruent to zero modulo p . Let I and L , be the degrees off and F,, respectively. It is clear that f and Fl can be chosen so that I L, < L. We can now bound the number of solutions (c,, . . 7) by considering the different values for x, in these solutions. Consider first those solutions for which f ( c n ) = 0 (mod p ) . 9) is fulfilled, then for any choice of cl, ... 7). 9) holds, is at most Ipn-'. Consider now solutions for which f(c,) $ 0 (mod p). All such solutions clearly satisfy the congruence Fl(x,, ...