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 rational field|, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.
This is the condition that it should be a subfield of where 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 a prime number, has a normal integral basis consisting of all the -th roots of unity other than. For a field contained in it, the field trace can be used to construct such a basis in also. Then in the case of squarefree and odd, is a compositum of subfields of this type for the primes dividing . This decomposition can be used to treat any of its subfields.
proved a converse to the Hilbert–Speiser theorem:
