Integral closure of a quartic extension

Abstract

Let R be a Noetherian unique factorization domain such that 2 and 3 are units, and let A = R[α] be a quartic extension over R by adding a root α of an irreducible quartic polynomial p(z) = z4 + az2 + bz + c over R. We will compute explicitly the integral closure of A in its fraction field, which is based on a proper factorization of the coefficients and the algebraic invariants of p(z). In fact, we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x) = 0. The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly. We compute also the ramifications of the integral closure over R.