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Let    p  ⩾  5    and   N   be prime numbers such that   p   divides    N  −  1   . We estimate the   p  ‐rank of the class group of    Q  (   N   1  /  p    )    in terms of the discrete logarithm, with values in    Z  /  p  Z   , of certain units. Using the Gross–Koblitz formula and identities on the   N  ‐adic Gamma function, we explicitly compute these logarithms. In particular, we give a new proof which does not use modular forms of a result of Calegari and Emerton.  Using the same method, we prove a special case of a twisted form of a conjecture of Gross about the relation between some Stickelberger element and the Galois structure of the class group of the cyclotomic field    Q  (   ζ  p   ,   ζ  N   )   .  A special case of our formulae, for which we do not have an elementary proof, is the following: Assume there are some integers   a  ,   b   such that    N  =   (    a  p   +   b  p    )   /   (   a  +  b   )    . Then(  a  +  b  )   ·   (   ∏   k  =  1    (   N  −  1   )  /  2k   8  k    )is a   p  th power modulo   N  .

On the Galois structure of the class group of certain Kummer extensions