2006/05/06 by Roland Quême, Roland Queme, Queme, Roland
Mathematics · #11R18 #11R29 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11R18 #msc:11R29
paper · pdf · doi:10.48550/arxiv.math/0605167
Some congruences for class number of quadratic fields and biquadratic fields contained in cyclotomic fields are derived of Stickelberger relation in section 7 p. 28
openalex publication_date 2006/05/06 · arxiv created 2006/07/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be an odd prime. Let Kp = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of Kp lying over p. Let G be the Galois group of Kp. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of Kp. Let P(sigma) = sigmap-2v-(p-2)+ ... + sigma v-1 +1 in Z[G], where vn is understood (mod p). We apply Stickelberger relation to odd prime numbers q different of p and to some singular integers A of Kp connected with the p-class group Cp of Kp and prove the pi-adic congruences: 1) pi2p-1 | AP(σ) if q = 1 (mod p), 2) pi2p-1 || AP(σ) if q = 1 (mod p) and p(q-1)/p = 1 (mod q). 3) pi2p | AP(σ) if q not = 1 (mod p). These results allow us to connect the structure of the p-class group Cp with pi-adic expression of singular numbers A and with solutions of some explicit congruences mod p in Z[X]. The last secion applies Stickelberger relation to describe the structure of the complete class group of Kp.