Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5

AbstractLet K/Q be a cyclic extension of degree l. Let ZK be the ring of integers of K. We say that ZK has a power basis (or is monogenic) if there exists θ ∈ ZK such that ZK = Z[θ]. We show that if l ≥ 5 is a prime, then ZK has no power basis, except in the well-known case where K is the maximal real subfield of a cyclotomic field; that is to say, if l is given, there exists, at most, one field K such that ZK has a power basis: the field Q0(2l + 1), when 2l + 1 is prime (e.g., for l = 5, ZK has a power basis only for the field K = Q0(11), and for l = 7, ZK never has a power basis)