of unity, and let Kn = Q(ζn) be the nth cyclotomic field. The discriminant of Kn 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 OK = Z[α], the discriminant of K is equal to the discriminant of the minimal polynomial of α. To see this, one can choose the integral basis of OK to be b1 = 1, b2 = α, b3 = α2, ..., bn = αn−1. Then, the matrix in the definition is the Vandermonde matrix associated to αi = σi(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 or , respectively. Basic results Brill's theorem: The sign of the discriminant is (−1)r2 where r2 is the number of complex places of K. A prime p ramifies in K if and only if p divides ΔK . Stickelberger's theorem: Minkowski's bound: Let n denote the degree of the extension K/Q and r2 the number of complex places of K, then Minkowski's theorem: If K is not Q, then |ΔK| > 1 (this follows directly from the Minkowski bound). Hermite–Minkowski theorem: Let N be a positive integer. There are only finitely many (up to isomorphisms) algebraic number fields K with |ΔK| < N. Again, this follows from the Minkowski bound together with Hermite's theorem (that there are only finitely many algebraic number fields with prescribed discriminant). History The definition of the discriminant of a general algebraic number field, K, was given by Dedekind in 1871. At this point, he already knew the relationship between the discriminant and ramification. Hermite's theorem predates the general