Quadratic residues and applications (original) (raw)

Let p be a prime number. We want to see when an element a of Z has a square root (mod p), i.e., in other words, we want to understand when √ a exists in Z p. A very useful notation to express this condition is the Legendre symbol a p whose value is 1 if a has a square root and is −1 in the other case (and is 0 if p divides a). The main result of the present thesis is Gauss's quadratic reciprocity law which implies that if p and q are two odd prime numbers, then: p q • q p = (−1) p−1 2 q−1 2 There are many different proofs of this theorem and we will analyze two of them which use quite different techniques. The thesis is organized as follows: in the first section we introduce some elementary constructions and the Berlekamp algorithm which allows one to compute a factorization of a polynomial in Z p [x]. Berlekamp theorems will turn out to be useful for proving Euler Criterion. Successively we give the two proofs of Gauss's reciprocity law mentioned above. Finally, in the last section, as a non trivial application of the theory developed, we give the Lucas-Lehmer theorem which allows to decide when a Mersenne number M p is prime.