In mathematics, the Hilbert–Speiser theorem is a result on

cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of ...

s, characterising those with a normal integral basis. More generally, it applies to any finite

abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...

of , which by the

Kronecker–Weber theorem In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q, having Galois group of the form (\mathbb Z/n\mathbb Z)^\times. The Kronecker–Weber theorem provides a partial conve ...

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 In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as ...

over . This is the condition that it should be a subfield of where is a

squarefree In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...

odd number. This result was introduced by in his

Zahlbericht In mathematics, the ''Zahlbericht'' (number report) was a report on algebraic number theory by . History In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski ...

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 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 (discrete Fourier tra ...

s. For example if we take a prime number , has a normal integral basis consisting of all the -th

roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...

other than . For a field contained in it, the

field trace In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' onto ''K''. Definition Let ''K'' be a field and ''L'' a finite extension (and hence a ...

can be used to construct such a basis in also (see the article on

s). Then in the case of squarefree and odd, is a

compositum In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...

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

of a fixed

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

has a relative normal integral basis if and only if . There is an elliptic analogue of the theorem proven by . It is now called the Srivastav-Taylor theorem .

References

* * * * * * {{DEFAULTSORT:Hilbert-Speiser theorem Cyclotomic fields Theorems in algebraic number theory