On the conjectures of Gross and Leopoldt

We show that the Leopoldt conjecture implies the so-called Hilbert’s theorem 90 for compact modules made from the $p$-units over the cyclotomic $\mathbb{Z}_p$-extension $k^{\mathrm{cyc}}_{\infty}$ of $k$. And under the generalized Gross conjecture, we show that Hilbert’s theorem 90 for the compact modules above is equivalent to the affirmation of the Leopoldt conjecture.

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Accepted 13 August 2014

Published 13 April 2016