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Generic cyclic polynomials of odd degree. (English) Zbl 0747.12003

A polynomial which parametrizes all Galois extensions with a given finite group \(G\) as the Galois group of overfields of a field \(K\) will be called a generic polynomial over \(K\) for the group \(G\). Thus \(X^ p-T\) \((X^ p-X-T)\) parametrizes cyclic extensions of degree \(p\) over fields of characteristic different from \(p\) (equal to \(p\)). In this interesting paper the author constructs explicitly generic polynomials for cyclic extensions of degree \(p^ n\) where \(p\) is an odd prime. This generic polynomial involves \(\varphi(p^ n)+1\) indeterminates and requires the \(p^ n\)-th roots of unity. The proof of the main theorem although straightforward, shows considerable ingenuity and involves several concepts.

MSC:

12F10 Separable extensions, Galois theory
11R32 Galois theory
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References:

[1] Frank DeMeyer, Separable Algebras over Commutative Rings (1971) · Zbl 0215.36602
[2] DOI: 10.1016/0022-314X(80)90075-X · Zbl 0428.12002
[3] Hardy >K., Carleton-Ottawa 7
[4] DOI: 10.1007/BF01389732 · Zbl 0292.20010
[5] DOI: 10.1016/0001-8708(82)90036-6 · Zbl 0484.12004
[6] Ward Smith Gene, Generic Cyclic Polynomials and Some Applications, PhD thesis (1990) · Zbl 0747.12003
[7] Washington Lawrence C., ”Introduction to Cyclotomic Fields” (1980) · Zbl 0966.11047
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