We consider the indices of subfields of cyclotomic ℤₚ-extensions of number fields. For the nth layer Kₙ of the cyclotomic ℤₚ-extension of ℚ, we find that the prime factors of the index of Kₙ/ℚ are those primes less than the extension degree pⁿ which split completely in Kₙ. Namely, the prime factor q satisfies , and this leads us to consider higher degree Fermat quotients. Indices of subfields of cyclotomic ℤₚ-extensions of a number field which is cyclic over ℚ with extension degree a prime different from p are also considered.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-2-1,
author = {Yoko Inoue and Kaori Ota},
title = {Indices of subfields of cyclotomic Zp-extensions and higher degree Fermat quotients},
journal = {Acta Arithmetica},
volume = {168},
year = {2015},
pages = {101-114},
zbl = {1334.11082},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-2-1}
}
Yoko Inoue; Kaori Ota. Indices of subfields of cyclotomic ℤₚ-extensions and higher degree Fermat quotients. Acta Arithmetica, Tome 168 (2015) pp. 101-114. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa169-2-1/