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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 , which by the Kronecker–Weber theorem are isomorphic to subfields of cyclotomic fields. :Hilbert–Speiser Theorem. A finite abelian extension has a normal integral basis if and only if it is tamely ramified over . 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 (see the article on Gaussian periods). Then in the case of squarefree and odd, is a compositum of subfields of this type for the primes dividing (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 of a fixed number field has a relative normal integral basis if and only if . ==References== 〔 * * * * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert–Speiser theorem」の詳細全文を読む スポンサード リンク
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