On the ideal class group of the normal closure of \(\mathbb{Q}(\sqrt[p]{n})\)
Fix a prime \(p>2\) and let \(\zeta_p\) be a primitive \(p\)-th root of unity. Building on previous work of Honda, Washington, Schoof, Hubbard-Washington, the author studies the ideal class group of the field \(K =\mathbb{Q}(\zeta_p, \sqrt[p]{n})\) where \(n\) is not a \(p\)-th power. He plans to study the Galois module structure of the class group \(\mathcal{C}\ell(K)\) of the field \(K\). For the non \(p\)-part of \(\mathcal{C}\ell(K)\), he uses Morita theory. For the \(p\)-part, he needs to assume that \(p\) does not divide the class number of \(\mathbb{Q}(\zeta_p)\) and that all primes \(\ell\) different from \(p\) that ramify in \(K\) are primitive roots modulo \(p\). An important result of the paper is the following one: Call \(\mathcal{C}\ell^{\,0}\) the kernel of the norm map from the class group of \(K\) to the class group of \(\mathbb{Q}(\zeta_p)\). Then there exist \(p-1\) copies of a finite abelian group \(H\) and a \(F_p\)-vector space \(V\) of dimension at most \({((p-3)/2)}^2\) such that the sequence \(\;0\longrightarrow V \longrightarrow \mathcal{C}\ell^{\,0} \longrightarrow H\times H\times\cdots \times H \longrightarrow 0\) is exact. The author brilliantly invoves the Galois cohomology groups machinery to achieve his goals.
scientific article; zbMATH DE number 7226983
scientific article