Indices of subfields of cyclotomic ${\mathbb Z}_p$-extensions and higher degree Fermat quotients
Volume 169 / 2015
Acta Arithmetica 169 (2015), 101-114
MSC: Primary 11R04; Secondary 11A07, 11R99.
DOI: 10.4064/aa169-2-1
Abstract
We consider the indices of subfields of cyclotomic ${\mathbb Z}_p$-extensions of number fields. For the $n$th layer $K_n$ of the cyclotomic ${\mathbb Z}_p$-extension of ${\mathbb Q}$, we find that the prime factors of the index of $K_n/{\mathbb Q}$ are those primes less than the extension degree $p^n$ which split completely in $K_n$. Namely, the prime factor $q$ satisfies $q^{p-1}\equiv 1 ({\rm mod} p^{n+1})$, and this leads us to consider higher degree Fermat quotients. Indices of subfields of cyclotomic ${\mathbb Z}_p$-extensions of a number field which is cyclic over ${\mathbb Q}$ with extension degree a prime different from $p$ are also considered.
