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Polynomials with Certain Prescribed Conditions on the Galois Group

Published online by Cambridge University Press:  20 November 2018

Elizabeth Rowlinson  and
Hans Schwerdtfeger
Elizabeth Rowlinson
Affiliation:
McGill University, Montreal, P.Q.
Hans Schwerdtfeger
Affiliation:
McGill University, Montreal, P.Q.
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In this paper, some contributions are made to the theory of algebraic equations over the rational field with conditions imposed on the Galois group. In § 1, for a given abstract group G all faithful permutation representations Ḡ are obtained, and it is shown that if one of them is the group of some equation with splitting field K, then any of them is the group of some equation, also with splitting field K. The method of proof enables us to construct an equation having as group a given faithful permutation representation Ḡ of a prescribed group G if we are given an equation having as group some faithful representation of G. In § 2, equations having nilpotent group are considered, non-normal extension fields are discussed, and a canonical form is obtained for the roots of non-normal irreducible equations; this form is used to characterize fields and equations with nilpotent groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 262 - 273
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Artin, E., Galois theory, 2nd ed. (Notre Dame Mathematical Lectures, no. 2, University of Notre Dame, Notre Dame, Indiana, 1959).Google Scholar
2. Cebotarev (Tschebotarôw), N. and Schwerdtfeger, H., Grundzùge der Galois'schen Théorie (P. Noordhoff, Groningen, 1950).Google Scholar
3. Hall, Marshall, Jr., The theory of groups (Macmillan, New York, 1959).Google Scholar
4. Loewy, A., Neue Elementare Begrundung und Erweiterung der Galois1 schen Théorie, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. 1925 (7), 1-50 and 1927 (1), 1–27.Google Scholar
5. van der Waerden, B. L., Modern algebra, Vol. I (Ungar, New York, 1953).Google Scholar
6. Young, L. M., On certain cyclic extensions of the field of rational numbers (Applied Mathematics and Statistical Laboratories, Stanford University, Technical Report, No. 1, 1961).Google Scholar