Reduction Modulo p

In this chapter, we will provide methods for studying Galois groups of monic polynomials P with integer coefficients by reduction modulo p (p prime number), that is using the polynomial \(\overline {P} \in {\mathbf {F}}_p[X]\) obtained by reduction modulo p of the coefficients of P. This will allow us to reduce to the situation of the field \({\mathbf {F}}_p\) , which is, at least theoretically, simpler: for example, we know how to factorize polynomials into irreducible polynomials (see Sect. 5.3 , Berlekamp’s algorithm (Fig. 10.1)), the extensions are always Galois with cyclic Galois groups, and we have the bonus of a canonical generator, the Frobenius morphism.