Hilbert–Speiser theorem

Hilbert–Speiser theorem

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 abelian extension "K" of the rational field "Q". The Kronecker–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 a subfield of

:"Q"(ζ"n")

where "n" is a squarefree odd number. This result is named for David Hilbert [It is Satz 132 of Hilbert's "Zahlbericht"; see Franz Lemmermeyer, "Reciprocity Laws: From Euler to Eisenstein" (2000), p. 388.] and Andreas 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 periods. 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, 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"(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


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