Editing Hilbert–Speiser theorem (section)

{{short description|Result on cyclotomic fields, characterising those with a normal integral basis}}
In [[mathematics]], the '''Hilbert–Speiser theorem''' is a result on [[cyclotomic field]]s, characterising those with a [[normal integral basis]]. More generally, it applies to any finite [[abelian extension]] of {{math|[[rational field|'''Q''']]}}, which by the [[Kronecker–Weber theorem]] are isomorphic to subfields of cyclotomic fields.

:'''Hilbert–Speiser Theorem.''' A finite abelian extension {{math|''K''/'''Q'''}} has a normal integral basis if and only if it is [[tamely ramified]] over {{math|'''Q'''}}.

This is the condition that it should be a [[Field extension|subfield]] of {{math|'''Q'''(''ζ<sub>n</sub>'')}} where {{mvar|n}} is a [[Square-free integer|squarefree]] [[odd number]]. This result was introduced by  {{harvs|txt|authorlink=David Hilbert|last=Hilbert|year1=1897|loc1=Satz 132|year2=1998|loc2=theorem 132}} in his [[Zahlbericht]] and by {{harvs|txt|authorlink=Andreas Speiser|last=Speiser|year=1916|loc=corollary to proposition 8.1}}.

In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of [[Gaussian period]]s. For example if we take {{mvar|n}} a prime number {{math|''p'' > 2}}, {{math|'''Q'''(''ζ<sub>p</sub>'')}} has a normal integral basis consisting of all the {{math|''p''}}-th [[roots of unity]] other than {{math|1}}. For a field {{mvar|K}} contained in it, the [[field trace]] can be used to construct such a basis in {{mvar|K}} also (see the article on [[Gaussian period]]s). Then in the case of {{mvar|n}} squarefree and odd, {{math|'''Q'''(''ζ<sub>n</sub>'')}} is a [[compositum]] of subfields of this type for the primes {{mvar|p}} dividing {{mvar|n}} (this follows from a simple argument on ramification). This decomposition can be used to treat any of its subfields.

{{harvs|txt | last1=Greither | first1=Cornelius | author1-link=Cornelius Greither | last2=Replogle | first2=Daniel R. | last3=Rubin | first3=Karl | author3-link=Karl Rubin | last4=Srivastav | first4=Anupam |year=1999}} proved a converse to the Hilbert–Speiser theorem:

:Each finite [[tamely ramified]] [[abelian extension]] {{mvar|K}} of a fixed [[number field]] {{mvar|J}} has a relative normal integral basis if and only if {{math|''J'' {{=}}'''Q'''}}.

There is an elliptic analogue of the theorem proven by  {{harvs|txt | last1=Srivastav | first1=Anupam  | last2=Taylor | first2=Martin J. | author2-link= Martin J. Taylor |year=1990}}. 
It is now called the Srivastav-Taylor theorem {{harvs|txt |year=1996}}.