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 of Q, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.

This result was introduced by Hilbert (1897, Satz 132, 1998, theorem 132) in his Zahlbericht and by Speiser (1916, corollary to proposition 8.1).

For example if we take n a prime number p > 2, Q(ζp) has a normal integral basis consisting of all the p-th roots of unity other than 1.

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).