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The minimum discriminants of quintic fields

Published online by Cambridge University Press:  18 May 2009

John Hunter
John Hunter
Affiliation:
The University Glasgow
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Let D be the discriminant of an algebraic number field F of degree n over the rational field R. The problem of finding the lowest absolute value of D as F varies over all fields of degree n with a given number of real (and consequently of imaginary) conjugate fields has not yet been solved in general. The only precise results so far given are those for n = 2, 3 and 4. The case n = 2 is trivial; n = 3 was solved in 1896 by Furtwangler, and n = 4 in 1929 by J. Mayer [6]. Reference to Furtwangler's work is given hi Mayer's paper. In this paper the results for n = 5, that is, for quintic fields, are obtained.

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 3 , Issue 2 , June 1957, pp. 57 - 67
Copyright
Copyright © Glasgow Mathematical Journal Trust 1957

References

REFERENCES

1.Berwick, W. E. H., Integral bases, Cambridge Tracts on Mathematics (no. 22) (Cambridge, 1927).Google Scholar
2.Cohn, H., A numerical study of quintics of small discriminant, Comm. Pure Appl. Math. 8 (1955), 377–386.CrossRefGoogle Scholar
3.Hardy, G. H., Littlewood, J. E., Polya, G., Inequalities (Cambridge, 1934), 52.Google Scholar
4.Hensel, K., Theorie der algebraischen Zahlen (Leipzig und Berlin, 1908), (a) 117, (b) 226, 231.Google Scholar
5.Hermite, C., Sur l'équation du cinquième degré, C. R. Acad. Sci. Paris, 62 (1866), (a) 67, (b) 71.Google Scholar
6.Mayer, J., Die absolut-kleinsten Diskriminanten der biquadratischen Zahlkörper, S.-B. Akad. Wiss. Wien Abt. Ila. 138 (1929), 733–742.Google Scholar