Hilbert-Speiser theorem
Encyclopedia
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
K of the rational field Q, which by the Kronecker–Weber theorem
are isomorphic to subfields of cyclotomic fields.
The Hilbert–Speiser theorem states that K has a normal integral basis if and only if it tamely ramified over Q. This is the condition that it should be a subfield of
where n 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 period
s. For example if we take n a prime number p > 2,
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 period
s). Then in the case of n squarefree and odd,
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.
proved
a converse to the Hilbert-Speiser theorem, stating that each finite tamely ramified abelian extension
K of a fixed
number field J has a relative normal integral basis if and only if J is the rational field Q.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Hilbert–Speiser theorem is a result on cyclotomic field
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers...
s, characterising those with a normal integral basis. More generally, it applies to any finite abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
K of the rational field Q, which by the Kronecker–Weber theorem
Kronecker–Weber theorem
In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words, every algebraic number field whose Galois group over Q is abelian, is a subfield of a cyclotomic field, i.e. a field obtained by adjoining a root...
are isomorphic to subfields of cyclotomic fields.
The Hilbert–Speiser theorem states that K has a normal integral basis if and only if it tamely ramified over Q. This is the condition that it should be a subfield of
- Q(ζn)
where n is a squarefree
Square-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
odd number. This result was introduced by in his Zahlbericht
Zahlbericht
In mathematics, the Zahlbericht was a report on algebraic number theory by .-History: and and the English introduction to give detailed discussions of the history and influence of Hilbert's 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 period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s. For example if we take n a prime number p > 2,
- Q(ζp)
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
Field trace
In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K...
can be used to construct such a basis in K also (see the article on Gaussian period
Gaussian period
In mathematics, in the area of number theory, a Gaussian period is a certain kind of sum of roots of unity. The periods permit explicit calculations in cyclotomic fields connected with Galois theory and with harmonic analysis . They are basic in the classical theory called cyclotomy...
s). Then in the case of n squarefree and odd,
- Q(ζn)
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.
proved
a converse to the Hilbert-Speiser theorem, stating that each finite tamely ramified abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
K of a fixed
number field J has a relative normal integral basis if and only if J is the rational field Q.