| 词条 | Hilbert–Speiser theorem |
| 释义 |
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 {{math|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 subfield of {{math|Q(ζn)}} where {{mvar|n}} is a 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 periods. For example if we take {{mvar|n}} a prime number {{math|p > 2}}, {{math|Q(ζp)}} 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 periods). Then in the case of {{mvar|n}} squarefree and odd, {{math|Q(ζn)}} 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 | last2=Replogle | first2=Daniel R. | last3=Rubin | first3=Karl | 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}}. References*{{Citation | last1=Greither | first1=Cornelius | last2=Replogle | first2=Daniel R. | last3=Rubin | first3=Karl | last4=Srivastav | first4=Anupam | title=Swan modules and Hilbert–Speiser number fields | journal=Journal of Number Theory | volume=79 | pages=164–173 | doi=10.1006/jnth.1999.2425 | year=1999}}
2 : Cyclotomic fields|Theorems in algebraic number theory |
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