THE FACTORIZATION OF 2 IN CUBIC FIELDS WITH INDEX 2

Abstract. We give the explicit factorization of the principal ideal < 2> in cubic fields with index 2. Let K be an algebraic number field. Let OK denote the ring of integers of K. Let d(K) denote the discriminant of K. Let θ ∈ OK be such that K = Q(θ). The minimal polynomial of θ over Q is denoted by irr Q (θ). The discriminant D(θ) and the index ind(θ) of θ are related by the equation (1) D(θ) = (ind(θ))2d(K). If p is a prime not dividing ind(θ) then it is well known that the following theorem of Dedekind gives explicitly the factorization of the principal ideal < p> of OK into prime ideals in terms of the irreducible factors of the min-imal polynomial irr Q (θ) modulo p, see for example [2, Theorem 10.5.1, p. 257]. Theorem 1. Let K = Q(θ) be an algebraic number field with θ ∈ OK. Let p be a rational prime. Let f(x) = ir