Quaternion extensions with restricted ramification

• Autorzy: Peter Schmid
• Języki publikacji: EN

Abstrakty

EN

In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime $p ≡ 1 (mod 2^{n-1})$, there exist unique real and complex normal number fields which are unramified outside S = {2,p} and cyclic over ℚ(√2) and whose Galois group is the (generalized) quaternion group $Q_{2^n}$ of order $2^n$.

Kategorie tematyczne

• 11F80: Galois representations
• 11R32: Galois theory
• 11S15: Ramification and extension theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2014
• Tom: 165
• Numer: 2
• Strony: 123-140

Daty

wydano

2014

Twórcy

autor

Peter Schmid

• Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Identyfikatory

DOI

10.4064/aa165-2-2