On the Galois structure of units of totally real $p$-rational fields

The Galois module structure of algebraic units is fundamental in number theory. However, its investigation is difficult because we need to understand arithmetic of number fields, and the integral representations of finite groups are difficult to classify. A number field is called $p$-rational if the Galois group of the maximal pro-$p$ $p$-ramified extension is free pro-$p$. The $p$-rationality is known to be a condition that reduces the complexities in problems in number theory. In this talk, we explain our results on the implication of the existing theories on integral representations of finite groups (factor equivalence, regulator constant, Yakovlev diagram) on the algebraic units of totally real $p$-rational fields. This talk is based on the joint works with Z. Bouazzaoui, D. Burns, A. Kumon, and C. Maire.