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A Characterization of One-Element p-Bases of Rings of Constants

Piotr Jędrzejewicz (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let K be a unique factorization domain of characteristic p > 0, and let f ∈ K[x₁,...,xₙ] be a polynomial not lying in K [ x p , . . . , x p ] . We prove that K [ x p , . . . , x p , f ] is the ring of constants of a K-derivation of K[x₁,...,xₙ] if and only if all the partial derivatives of f are relatively prime. The proof is based on a generalization of Freudenburg’s lemma to the case of polynomials over a unique factorization domain of arbitrary characteristic.

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...

Characteristic of Rings. Prime Fields

Christoph Schwarzweller, Artur Korniłowicz (2015)

Formalized Mathematics

The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.

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