## 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

### Keywords:

generic polynomial; cyclic extensions
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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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