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Quaternion extensions with restricted ramification

100%
Schmid P.
Acta Arithmetica
|
2014
|
tom 165
|
nr 2
123-140
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 $2^n$.
2
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On 2-extensions of the rationals with restricted ramification

100%
Schmid P.
Acta Arithmetica
|
2014
|
tom 163
|
nr 2
111-125
EN
For a finite group G let 𝒦₂(G) denote the set of normal number fields (within ℂ) with Galois group G which are 2-ramified, that is, unramified outside {2,∞}. We describe the 2-groups G for which 𝒦₂(G) ≠ ∅, and determine the fields in 𝒦₂(G) for certain distinguished 2-groups G appearing (dihedral, semidihedral, modular and semimodular groups). Our approach is based on Fröhlich's theory of central field extensions, and makes use of ring class field constructions (complex multiplication).
3
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The Stickelberger element of an imaginary quadratic field

100%
Schmid P.
Acta Arithmetica
|
1999
|
tom 91
|
nr 2
165-169
4
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Correction to "Computing Galois groups by means of Newton polygons" (Acta Arith. 115 (2004), 71-84)

64%
Kölle M., Schmid P.
Acta Arithmetica
|
2009
|
tom 140
|
nr 1
101-103
5
Content available remote

Computing Galois groups by means of Newton polygons

64%
Kölle M., Schmid P.
Acta Arithmetica
|
2004
|
tom 115
|
nr 1
71-84
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