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# Mathematical Research Letters

## Volume 22 (2015)

### Number 5

### On the conjectures of Gross and Leopoldt

Pages: 1509 – 1540

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n5.a11

#### Author

#### Abstract

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.