Construction of the integral closure of a finite integral domain. II

In a previous paper the problem of constructing the integral closure of a finite integral domain k\x.,..., x 1 = k[x] was considered. A reduction to the case dtk(x)/k = 1, k(x)/k separable, and zz = 2 was made. A subsidiary problem was: if k[x\ is not integrally closed, to find a y in k(x) integral over k[x\ but not in it. This was done for n = 2, but should have been done for arbitrary zz. The extra details are here given. For the convenience of the reader, the full argument is sketched. In [2] we proposed to construct the integral closure of a finite integral domain k[xl, • • • , x ] = k[x] in its quotient field &(*,, • • • , x ). Three subsidiary problems were formulated, of which the first two were: 1. to give a method for deciding whether k[x] is integrally closed; 2. in the case k[x] is not integrally closed, to give a method for finding an element in k(x) integral over k[x] but not in it. We dealt first with the case that k(x)/k is separable, and a reduction to the case degree of transcendency of k(x)/k = 1 was made. It is then easy to reduce the original problem to the case n = 2, but on p. 7 it was stated, though incorrectly, that the subsidiary problem 2 was thus reduced. The slip was (in effect) noted in [l]. This is a note of correction. Basically we assume a familiarity with [2], but, for the convenience of the reader, try to rely on [2] as little as possible. For another treatment (not quite complete) of the problems here considered see [5]. 1. Preliminaries. Reference [3] considers some basic construction problems in a polynomial ring k[X., • • • , X ] = k[X]. If A is an ideal in k[X], given via a finite basis, and b £ k[X] is given, one can decide whether b £ A (§5), find the dimension of A (§6), and construct A n k[X^, • • • , X _ j] (§23, Note 4; see also [2, p. 17]). Hence if dim A = 0, by contracting A to &[X.], one can find a polynomial whose roots are precisely the z'th coordinates of the points annihilating A. If dim A = r and A = Q.Ci •• ■ O Q is a normal decomposition of A into primary ideals, one can construct the intersection of the r-dimensional Q. (§17). Given ideals A and B, one can construct Received by the editors November 13, 1973. AMS (MOS) subject classifications (1970). Primary 02E99, 13F20, 13B20.