On the ring of p-integers of a cyclic p-extension over a number field

Let p be a prime number. A finite Galois extension N/F of a number field F with group G has a normal p-integral basis (p-NIB for short) when O′ N is free of rank one over the group ring O′ F [G]. Here, O′ F = OF [1/p] is the ring of p-integers of F . Let m = p be a power of p and N/F a cyclic extension of degree m. When ζm ∈ F×, we give a necessary and sufficient condition for N/F to have a p-NIB (Theorem 3). When ζm 6∈ F× and p [F (ζm) : F ], we show that N/F has a p-NIB if and only if N(ζm)/F (ζm) has a p-NIB (Theorem 1). When p divides [F (ζm) : F ], we show that this descent property does not hold in general (Theorem 2). Humio Ichimura Faculty of Science Ibaraki University 2-1-1, Bunkyo, Mito, Ibaraki, 310-8512 Japan E-mail : [email protected] Manuscrit reçu le 4 septembre 2003.