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Indices of subfields of cyclotomic ℤₚ-extensions and higher degree Fermat quotients

Yoko Inoue, Kaori Ota (2015)

Acta Arithmetica

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 q p - 1 1 ( m o d p n + 1 ) , 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...

Isomorphisms of algebraic number fields

Mark van Hoeij, Vivek Pal (2012)

Journal de Théorie des Nombres de Bordeaux

Let ( α ) and ( β ) be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, ( β ) ( α ) . The algorithm is particularly efficient if there is only one isomorphism.

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