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A generalization of a result on integers in metacyclic extensions

James Carter (1999)

Colloquium Mathematicae

Let p be an odd prime and let c be an integer such that c>1 and c divides p-1. Let G be a metacyclic group of order pc and let k be a field such that pc is prime to the characteristic of k. Assume that k contains a primitive pcth root of unity. We first characterize the normal extensions L/k with Galois group isomorphic to G when p and c satisfy a certain condition. Then we apply our characterization to the case in which k is an algebraic number field with ring of integers ℴ, and, assuming some...

A monogenic Hasse-Arf theorem

James Borger (2004)

Journal de Théorie des Nombres de Bordeaux

I extend the Hasse–Arf theorem from residually separable extensions of complete discrete valuation rings to monogenic extensions.

A propos de la relation galoisienne x 1 = x 2 + x 3

Franck Lalande (2010)

Journal de Théorie des Nombres de Bordeaux

L’existence d’un polynôme f , irréductible sur un corps k de caractéristique 0 et dont trois racines vérifient la relation linéaire x 1 = x 2 + x 3 , ne dépend que de la paire de groupes finis ( G , H ) G = Gal k ( f ) et H G est le fixateur d’une racine. Le cas régulier ( H = 1 ) est désormais assez bien décrit. On démontre dans ce texte que pour de nombreuses paires ( G , H ) primitives ( H sous-groupe maximal de G ) et en particulier pour toutes celles de degré 50 , la relation x 1 = x 2 + x 3 n’est pas réalisable.En appendice, Joseph Oesterlé démontre que cette...

A really elementary proof of real Lüroth's theorem.

T. Recio, J. R. Sendra (1997)

Revista Matemática de la Universidad Complutense de Madrid

Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if K is a subfield...

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