Indépendance linéaire sur \=Q de logarithmes p-adiques de nombres algébriques et rang p-adique du groupe des unités d'un corps de nombres
Let k be a Galois extension of \({\mathbb{Q}}\) (resp. of an imaginary quadratic field \(k_ 0)\) and \(G=Gal(k/{\mathbb{Q}})\) (resp. \(G=Gal(k/k_ 0))\). Let E be the group of units of k, p a prime, and \(r_ p\) the p-adic rank of E. Finally denote by S the set of irreducible characters of G which occur in the decomposition of \(E\otimes {\bar {\mathbb{Q}}}\). Using Ax's method [\textit{J. Ax}, Ill. J. Math. 9, 584-589 (1965; Zbl 0132.283)], the authors deduce the lower bound \(r_ p=\sum_{\chi\in S}\deg \chi\) from the p- adic version of Baker's theorem on linear forms in logarithms [\textit{A. Brumer}, Mathematika 14, 121-124 (1967; Zbl 0171.011)]. As a corollary, they prove Leopoldt's conjecture for imaginary Galois extensions of \({\mathbb{Q}}\) with Galois group \(A_ 4\). They also give a lower bound for the p-adic rank of the p-units of k.
lower bound for the p-adic rank of the p-units
scientific article; zbMATH DE number 3871495