- Hilbert–Speiser theorem
In
mathematics , the Hilbert–Speiser theorem is a result oncyclotomic field s, characterising those with anormal integral basis . More generally, it applies to anyabelian extension "K" of therational field "Q". TheKronecker–Weber theorem characterises such "K" as (up to isomorphism ) the subfields of:"Q"(ζ"n")
where
:ζ"n" = "e"2π"i"/"n".
In abstract terms, the result states that "K" has a normal integral basis if and only if it
tamely ramified over "Q". In concrete terms, this is the condition that it should be asubfield of:"Q"(ζ"n")
where "n" is a squarefree
odd number . This result is named forDavid Hilbert [It is Satz 132 of Hilbert's "Zahlbericht"; see Franz Lemmermeyer, "Reciprocity Laws: From Euler to Eisenstein" (2000), p. 388.] andAndreas Speiser 1885-1970.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 "n" a prime number "p" > 2,:"Q"(ζ"p")
has a normal integral basis consisting of the "p" − 1 "p"-th
roots of unity other than 1. For a field "K" contained in it, thefield trace can be used to construct such a basis in "K" also (see the article onGaussian period s). Then in the case of "n" squarefree and odd,:"Q"(ζ"n")
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.Notes
Wikimedia Foundation. 2010.