Discriminant of An Algebraic Number Field - Examples

Examples

• Quadratic number fields: let d be a square-free integer, then the discriminant of is

An integer that occurs as the discriminant of a quadratic number field is called a fundamental discriminant.

• Cyclotomic fields: let n > 2 be an integer, let ζ_n be a primitive nth root of unity, and let K_n = Q(ζ_n) be the nth cyclotomic field. The discriminant of K_n is given by

where is Euler's totient function, and the product in the denominator is over primes p dividing n.

• Power bases: In the case where the ring of integers has a power integral basis, that is, can be written as O_K = Z, the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can chose the integral basis of O_K to be b₁ = 1, b₂ = α, b₃ = α2, ..., b_n = αn−1. Then, the matrix in the definition is the Vandermonde matrix associated to α_i = σ_i(α), whose determinant squared is

which is exactly the definition of the discriminant of the minimal polynomial.

• Let K = Q(α) be the number field obtained by adjoining a root α of the polynomial x3 − x2 − 2x − 8. This is Richard Dedekind's original example of a number field whose ring of integers does not possess a power basis. An integral basis is given by {1, α, α(α + 1)/2} and the discriminant of K is −503.
• Repeated discriminants: the discriminant of a quadratic field uniquely identifies it, but this is not true, in general, for higher-degree number fields. For example, there are two non-isomorphic cubic fields of discriminant 3969. They are obtained by adjoining a root of the polynomial x3 − 21x + 28 or x3 − 21x − 35, respectively.