Hilbert–Speiser theorem

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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 abelian extension K of the rational field Q. The Kronecker–Weber theorem characterises such K as (up to isomorphism) the subfields of

Q(ζ_n)

where

ζ_n = e^2πi/n.

In abstract terms, the result states that K has a normal integral basis if and only if it tamely ramified over Q. In concrete terms, this is the condition that it should be a subfield of

Q(ζ_n)

where n is a squarefree odd number. This result is named for David Hilbert^[1] and Andreas Speiser 1885-1970.

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(ζ_p)

has a normal integral basis consisting of the p − 1 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(ζ_n)

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.

[edit] Notes

1. ^ It is Satz 132 of Hilbert's Zahlbericht; see Franz Lemmermeyer, Reciprocity Laws: From Euler to Eisenstein (2000), p. 388.