Hilbert–Speiser theorem

In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension of Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.

Hilbert–Speiser Theorem. A finite abelian extension K/Q has a normal integral basis if and only if it is tamely ramified over Q.

This is the condition that it should be a subfield of Q(ζn) where n is a squarefree odd number. This result was introduced by Hilbert (1897,Satz 132, 1998,theorem 132) in his Zahlbericht and by Speiser (1916,corollary to proposition 8.1).

In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods. For example if we take n a prime number p > 2, Q(ζp) has a normal integral basis consisting of all the p-th roots of unity other than 1. For a field K contained in it, the field trace can be used to construct such a basis in K also (see the article on Gaussian periods). Then in the case of n squarefree and odd, Q(ζn) is a compositum of subfields of this type for the primes p dividing n (this follows from a simple argument on ramification). This decomposition can be used to treat any of its subfields.

Cornelius Greither, Daniel R. Replogle, and Karl Rubin et al. (1999) proved a converse to the Hilbert–Speiser theorem:

Each finite tamely ramified abelian extension K of a fixed number field J has a relative normal integral basis if and only if J =Q.

References

    • Greither, Cornelius; Replogle, Daniel R.; Rubin, Karl; Srivastav, Anupam (1999), "Swan modules and Hilbert–Speiser number fields", Journal of Number Theory, 79: 164–173, doi:10.1006/jnth.1999.2425
    • Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456
    • Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62779-1, MR 1646901
    • Speiser, A. (1916), "Gruppendeterminante und Körperdiskriminante", Mathematische Annalen, 77 (4): 546–562, doi:10.1007/BF01456968, ISSN 0025-5831
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