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 K of the rational field Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields. The Hilbert–Speiser theorem states that K 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 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 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(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, stating that each finite tamely ramified abelian extension K of a fixed number field J has a relative normal integral basis if and only if J is the rational field Q.