3 Added "(for some even $k\in[2,p-3]$)".; added 3 characters in body
Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$ . (for some even $k\in[2,p-3]$). This implies that the class number of the field $K$ of $p$-th roots of unity is divisible by $p$. Let $L$ be the field of $p^2$-th roots of unity. What, if anything, is known on about the capitulation of ideal classes in $L/K$ ( we say that an ideal class from $K$ capitulates in $L$ if an ideal generating this class becomes principal there)? It is possible to write down criteria in terms of units that are or are not norms from $L$, but this does not seem to help a lot. I am mainly interested in the question whether there is a connection between the index $k$ and the capitulation of the subgroup of order $p$ corresponding to $k$ via Herbrand-Ribet. I am pretty sure that classical algebraic number theorists did not do an awful lot in this direction but I am not familiar with any advances in Iwasawa theory: whether an ideal class capitulates in $L/K$ is encoded in the Hilbert class field, so the structure of the maximal abelian unramified $p$-extension of the cyclotomic Iwasawa extension of $K$ might contain relevant information. Does it?
Let $p$ be an irregular prime, which means that $p$ divides some Bernoulli number: $p \mid B_k$. This implies that the class number of the field $K$ of $p$-th roots of unity is divisible by $p$. Let $L$ be the field of $p^2$-th roots of unity. What, if anything, is known on the capitulation of ideal classes in $L/K$ ( we say that an ideal class from $K$ capitulates in $L$ if an ideal generating this class becomes principal there)? It is possible to write down criteria in terms of units that are or are not norms from $L$, but this does not seem to help a lot. I am mainly interested in the question whether there is a connection between the index $k$ and the capitulation of the subgroup of order $p$ corresponding to $k$ via Herbrand-Ribet. I am pretty sure that classical algebraic number theorists did not do an awful lot in this direction but I am not familiar with any advances in Iwasawa theory: whether an ideal class capitulates in $L/K$ is encoded in the Hilbert class field, so the structure of the maximal abelian unramified $p$-extension of the cyclotomic Iwasawa extension of $K$ might contain relevant information. Does it?