Hilbert–Speiser theorem

Template:Short description 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 Template:Math, which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields.

Hilbert–Speiser Theorem. A finite abelian extension Template:Math has a normal integral basis if and only if it is tamely ramified over Template:Math.

This is the condition that it should be a subfield of Template:Math where Template:Mvar is a squarefree odd number. This result was introduced by Template:Harvs in his Zahlbericht and by Template:Harvs.

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 Template:Mvar a prime number Template:Math, Template:Math has a normal integral basis consisting of all the Template:Math-th roots of unity other than Template:Math. For a field Template:Mvar contained in it, the field trace can be used to construct such a basis in Template:Mvar also (see the article on Gaussian periods). Then in the case of Template:Mvar squarefree and odd, Template:Math is a compositum of subfields of this type for the primes Template:Mvar dividing Template:Mvar (this follows from a simple argument on ramification). This decomposition can be used to treat any of its subfields.

Template:Harvs proved a converse to the Hilbert–Speiser theorem:

Each finite tamely ramified abelian extension Template:Mvar of a fixed number field Template:Mvar has a relative normal integral basis if and only if Template:Math.

There is an elliptic analogue of the theorem proven by Template:Harvs. It is now called the Srivastav-Taylor theorem Template:Harvs.

References

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