Nicolae Popescu

Invertible and divisorial ideals of generalized Dedekind domains

Abstract In this paper we give an ideal-theoretical characterization of a distinguished class of Prufer domains, the class of generalized Dedekind domains. Namely, we prove that a Prufer domain R is generalized Dedekind if and only if the divisorial ideals of R are exactly the ideals of type JP l … P n , where J is an invertible fractional ideal and P l , …, P n are (incomparable) nonzero prime ideals of R . We also show that, when R is a generalized Dedekind domain, the group of fractional invertible ideals of R is isomorphic to the free abelian group generated by the set of nonzero prime ideals of R and a basis for it is given by a suitable set of two-generated ideals with prime radical.

A GALOIS THEORY FOR THE FIELD EXTENSION K (( X ))/ K

Let K be a field of characteristic 0, which is algebraically closed to radicals. Let F = K (( X )) be the valued field of Laurent power series and let G = Aut ( F / K ). We prove that if L is a subfield of F , K ≠ L , such that L / K is a sub-extension of F / K and F / L is a Galois algebraic extension ( L / K is Galois coalgebraic in F / K ), then L is closed in F , F / L is a finite extension and Gal ( F / L ) is a finite cyclic group of G . We also prove that there is a one-to-one and onto correspondence between the set of all finite subgroups of G and the set of all Galois coalgebraic sub-extensions of F / K . Some other auxiliary results which are useful by their own are given.

Sur la structure des anneaux absoluments plats commutatifs

Les anneaux absolument plats commutatifs (ou en abrg il est bien connu que son spectre (voir [2]) est un espace topologique &park et totalement discontinu (espace de Book) et les anneaux locaux associks sont des corps. I1 sembleratit done a premikre vue, qu’un anneau a.p. est bien dktermini: par un espace de Boole et une famille de corps. Cet article se propose d’essayer d’Clucider cette question, c’est-g-dire d’apprkcier en quelle mesure un anneau a.p. est dCterminC par son spectre et par les cwps rksiduels assock. x cet effet nous avons introduit dans la 3kme section la notion de systkme de corps, qui, selon notre opinion, est susceptible de contribuer 2 clarifier la structure des anneaux a.p. La notion de K-anneau qui a ktk introduite, ne se distingue que de manikre form& du concept courant d’un nnneau a.p. Le ThCorkme 3.4 exprime les conditions nkcessaircs et suffisantes pour I’existence des K-anneaux dans un systkmc de corps dktermink. La notion d’extension d’un corps est g&nCralisCe dans le cas des anneaus a.p. dans lc cadre de la Section 4. C’est 12 kgalement qui nous avons introduit

Spectral norms on valued fields

Abstract. Let

Galois structures on plane compacts

(K,\left| .\right| )

TOTAL VALUATION RINGS OF K(X, σ) CONTAINING K

be a perfect valued field,

Analytic normal basis theorem

\overline{K}

A Galois theory for the Banach algebra of continuous symmetric functions on absolute Galois groups

be an algebraic closure of

Trace series on

K,