Indices of subfields of cyclotomic ${\mathbb Z}_p$-extensions and higher degree Fermat quotients

Yoko Inoue¹ ; Kaori Ota¹

¹ Department of Mathematics Tsuda College 2-1-1 Tsuda-cho, Kodaira-shi, Tokyo 187-8577, Japan

Acta Arithmetica, Tome 169 (2015) no. 2, pp. 101-114

Résumé

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.

Détail

DOI : 10.4064/aa169-2-1

Keywords: consider indices subfields cyclotomic mathbb p extensions number fields nth layer cyclotomic mathbb p extension mathbb prime factors index mathbb those primes extension degree which split completely namely prime factor satisfies p equiv nbsp mod nbsp leads consider higher degree fermat quotients indices subfields cyclotomic mathbb p extensions number field which cyclic mathbb extension degree prime different considered