ON THE p-CLASS GROUP STABILITY ALONG CYCLIC p-TOWERS OF A NUMBER FIELD
Résumé
Let k be a number field and let p≥2 be a prime number. We call K/k a cyclic p-tower if Gal(K/k)=Z/p^N Z. We give an elementary proof of a stability theorem for generalized p-class groups X_n along the tower (Theorem 3.1).
Using generalizations of Chevalley's formula (Gras, J. Math. Soc. Japan 46(3) (1994), Proc. Math. Sci. 127(1) (2017)), we improve results by Fukuda (1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020) and others, whose techniques are based on Iwasawa's theory, pro-p-group Galois theory or specific methods. The main difference, compared to these works, is that we introduce a parameter lambda≥0 giving formulas of the form #X_n=#X_0 p^(lambda n), n in [0,N].
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