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(ζ<sub>n</sub>) where n is a squarefree odd number. This result was introduced by in his Zahlbericht and by .

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<sub>p</sub>) 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(ζ<sub>n</sub>) 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.

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.

There is an elliptic analogue of the Hilbert- Speiser theorem proven by . 

It is now called the Srivastav-Taylor theorem .