Integral closure of a quartic extension  

作　　者：TAN ShengLi XIE DaJun 

机构地区：[1]Department of Mathematics and Shanghai Key Laboratory of PMMP, East China Normal University

出　　处：《Science China Mathematics》2015年第3期553-564,共12页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11231003);the Science Foundation of Shanghai(Grant No.13DZ2260600);East China Normal University Reward for Excellent Doctors in Academics(Grant No.XRZZ2012014)

摘　　要：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 rootαof 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.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 rootαof an irreducible quartic polynomial p（z）=z^4＋az^2＋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 z^4＋a（x）z^2＋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.

关 键 词：algebraic invariants quartic extension integral closure DISCRIMINANT SYZYGY 

分 类 号：O156[理学—数学]