Discriminant of an algebraic number field

In algebraic number theory, the discriminant called a principal ideal in a whole ring, which makes a number-theoretic statement about the body expansion of a number field.

Definition

Be a ring, a subring such that a free module of rank is. For is called the discriminant of.

If is a basis of, the Diskriminate is determined up to a unit in clear, especially so the principal ideal generated by in is independent of the basis choice. This principal ideal is denoted by and is called the discriminant of over.

Properties and application

• Be a body of characteristic, a field extension of the degree and the different - Algebrenmonomorphismen of the algebraic degree. Then for a base of:

• Be two number fields, with the associated wholeness rings. Then for a prime ideal the following: if and only branched if applies. In particular, the fact that there are only finitely many prime ideals branched (unique prime factorization of, see Dedekind ring ) follows.

Example

Be; denote the equivalence class of in.

Thus, corresponding to the discriminant of the polynomial.

For the calculation of the tracks used in this case: