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

Volume 169 / 2015

Yoko Inoue, Kaori Ota 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.

Authors

  • Yoko InoueDepartment of Mathematics
    Tsuda College
    2-1-1 Tsuda-cho, Kodaira-shi, Tokyo 187-8577, Japan
  • Kaori OtaDepartment of Mathematics
    Tsuda College
    2-1-1 Tsuda-cho, Kodaira-shi, Tokyo 187-8577, Japan
    e-mail

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