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(ζn)
where
- ζn = e2πi/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(ζn)
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(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.
