A general Kummer theory for function fields

Lane, Saunders Mac; Schilling, O. F. G.

doi:10.1215/s0012-7094-42-00912-8

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…After the completion of this paper MacLane and the author discovered that analogous results can be developed for function fields. Some of the results obtained in[11] can be proved for relatively complete fields. We quote especially the theorems on the explicit structure of the possible Galois groups as group extensions.…”

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confidence: 89%

“…By a simple well-ordering argument we can suppose 12 For a proof 13 one has to observe that the elements of S define on every finite subfield K/F the same automorphisms as the elements of G (L/F). The 11 Since the property of an element being integral is transitive and since the prime ideal P of F is divisible by exactly one prime ideal of F, we agree to use the symbol P ambiguously for the prime ideals of the various algebraic extensions K D F. 12 For the Galois theory of infinite extensions see [3], [6] …”

Section: (1/n)b(na) the Finiteness Of The Degree N Implies That (I)mentioning

confidence: 99%

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Normal Extensions of Relatively Complete Fields

Schilling¹

1943

American Journal of Mathematics

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. This paper is a discussion of some relations between the Hilbert theory and the existence problem for normal extensions of relatively complete fields. The Hilbert theory describes the subfields of a given normal field by means of the arithmetic inherent in the fields in question. The results of this theory may be interpreted as necessary conditions for the existence of normal extensions over a base field. We propose the question whether such conditions are sufficient for the solution of existence problems. To answer this query it is necessary to construct explicitly normal extensions taking advantage of information obtained by the Hilbert theory. A. A. Albert, 0. Ore and the author 1 have investigated independently such problems for the classical fields of p-adic numbers. In this article we take general relatively complete fields as reference fields. A field is termed relatively complete if Hensel's Lemma holds for the prime ideal of non-units with respect to the given valuation.2 The normal extensions involved are mostly of infinite degree over the ground field. This generality leads to an understanding of the mechanism underlying the structure of normal extensions. We have purposely selected relatively complete fields rather than complete fields as base fields since relative completeness can always be established for infinite extensions.3In the first part of the paper we develop the Hilbert theory for arbitrary normal extensions L over a relatively complete field F with value group r and perfect residcue class fields 5. We prove that L contains a unique maximal unramified field LI whose residue class field coincides with that of L. The field L contains, relative to LI, a radical extension LR, the ramification field, which is determined by the structure of the value group r (L) of L. As a matter of fact, the value group r (LB) of LR consists of all those elements of F(L) whose orders modulo r are prime to the characteristic X of F. Moreover, the field LI contains all roots of unity whose orders occur as orders of the elements in the factor group r (LR) /r. In case L contains only elements whose degrees over F are prime to X we find L = LR. Thus normal extensions * Presented to the Society, Dec. 29-31, 1941. Received October 29, 1941.

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confidence: 89%

Section: (1/n)b(na) the Finiteness Of The Degree N Implies That (I)mentioning

confidence: 99%

Normal Extensions of Relatively Complete Fields

Schilling¹

1943

American Journal of Mathematics

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“…Introduction. The construction of valued fields extending a given one and having prescribed value groups and residue fields with respect to the extensions of the given valuation has received much attention in the literature from many points of view; see [l], [2], [3], bl, [6], [7], [8], [ll], [12], for example. The Let (K, A) be a (Krull-) valued field, ß a separable closure of K, and C an extension of A to ß.…”

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confidence: 99%