Hilbert–Speiser theorem

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title: "Hilbert–Speiser theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["cyclotomic-fields", "theorems-in-algebraic-number-theory"] description: "Result on cyclotomic fields, characterising those with a normal integral basis" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Hilbert–Speiser_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Result on cyclotomic fields, characterising those with a normal integral basis ::

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(ζn) 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(ζp) 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(ζn) 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 .

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

References

• {{Citation | last1=Agboola | first1=A. | title= Torsion points on elliptic curves and Galois module structure | year=1996 | journal=Invent Math | volume=123 | pages= 105–122 | doi=10.1007/BF01232369 | bibcode=1996InMat.123..105A }}
• {{Citation | last1=Srivastav | first1=Anupam |last2=Taylor | first2=Martin J. | title=Elliptic curves with complex multiplication and Galois module structure | year=1990 | journal=Invent Math | volume=99 | pages= 165–184 | doi=10.1007/BF01234415 | bibcode=1990InMat..99..165S }}

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