Discriminant of an algebraic number field

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In mathematics, the discriminant of an algebraic number field is a numerical invariant containing information about ramified primes.

[edit] Explanation

If K is an algebraic number field and O_K its ring of integers, the discriminant of K is associated to O_K and in some sense measures how large O_K is. In the special case of O_K = Z[α] for some algebraic integer α in K, it is simple to define, as the discriminant of the minimal polynomial P_α of α. This suffices, for example, in the case of the Gaussian integers: we take P(T) = T² + 1 for the choice α = i and calculate the discriminant as −4.

This in fact works for any quadratic field or cyclotomic field; but certainly not in general. There we can only be sure that Z[α] can be chosen to be of finite index in O_K as an abelian group. This gives a factor (of the index) which is awkward to apply. The correct definition comes through a recognition that the discriminant of a polynomial is a square of a Vandermonde determinant, and that determinant is what we should generalise. The analogue in the general case is this: let the ω_i be an integral basis (i.e. basis for O_K as Z-module) and form

det(ω_i^(j))

where the superscripts mean that we take the conjugates. This (squared) leads to the correct general definition.

Why this is the correct approach is best studied in terms of the real vector space K ⊗_Q R, and the embedding of O_K into it as a lattice. The determinant involved in the discriminant then has a simple interpretation as a volume of a fundamental region for O_K.

There is also a formula for the discriminant related to the quadratic form definition above, starting from the field trace. In the theory of Pontryagin duality for completions of K as local fields, the related different ideal occurs naturally in matching up Haar measures. This accounts for the role of the discriminant in the functional equation for the Dedekind zeta function, and thence in the analytic class number formula, and Brauer-Siegel theorem.

A theorem of Stickelberger states that the discriminant D of an algebraic number field must be congruent to 0 or 1 modulo 4. A result of Kronecker is that D = 1 is possible only for the rational number field Q; this entails that every other number field has some ramified prime p in it. Lower bounds for discriminants, in terms of the degree, are proved by methods from the geometry of numbers and analytic number theory. In the case of an abelian extension, one of the results of class field theory (conductor-discriminant formula) is a factorisation of the discriminant according to characters, which in the case of an abelian extension of Q are Dirichlet characters.