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 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 2π i / 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 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.