Semidirect imbedding problem with a metabelian kernel

Ishkhanov, V. V.

doi:10.1007/bf01885512

J Math Sci

1982

DOI: 10.1007/bf01885512

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Semidirect imbedding problem with a metabelian kernel

V. V. Ishkhanov¹

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1994

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“…reached a high point in the proof by Shafarevich [16] that every solvable group occurs as a Galois group over the rationals (cf. [10]). A related line was the study of "Galois algebras" initiated by Hasse [9] (after one earlier paper by Brauer [6]); an updated version of the relevant parts of this can be found in [14].…”

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confidence: 99%

The Normal Closures of Certain Kummer Extensions

Waterhouse

1994

Can. math. bull.

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Let F be a field containing a primitive p-th root of unity, let K / F be a cyclic extension with group 〈σ〉 of order pn, and choose a in K. This paper shows how the Galois group of the normal closure of K(a1/p) over F can be determined by computations within K. The key is to define a sequence by applying the operation x ↦ σ(x)/x repeatedly to a. The first appearance of a p-th power determines the degree of the extension and restricts the Galois group to one or two possibilities. A certain expression involving that p-th root and the terms in the sequence up to that point is a p-th root of unity, and the group is finally determined by testing whether that root is 1. When (σ(a)/a G Kp, the results reduce to a theorem of A. A. Albert on cyclic extensions.

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confidence: 99%

The Normal Closures of Certain Kummer Extensions

Waterhouse

1994

Can. math. bull.

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Semidirect imbedding problem with a metabelian kernel

Cited by 1 publication

References 1 publication

The Normal Closures of Certain Kummer Extensions

The Normal Closures of Certain Kummer Extensions

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