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.


 * 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 $Q(ζ_{n})$ 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 $p > 2$, $Q(ζ_{p})$ 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, $Q(ζ_{n})$ 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.

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 theorem proven by. It is now called the Srivastav-Taylor theorem.