Re: Another algebra question

On Sun, 30 Sep 2007 01:55:37 -0700, Snis Pilbor <snispilbor@xxxxxxxxx>
wrote:

Thanks for you guys' help on my previous algebra question. Here is
another thing I am stumped on. Suppose R is a ring in a field F, and
R is integrally closed in its field of fractions. I want to show that
an element x in F is integral over R (that is, it's annihilated by
some monic polynomial with coefficients in R), iff its minimal
polynomial has coefficients in R.

When I say R is integrally closed in its field of fractions, I mean:
if z is in the field of fractions, and z is killed by a monic
polynomial with coefficients in R, then z is in R.

Obviously the minimal polynomial divides any other polynomial which
kills x. Unfortunately this doesn't seem to be enough. For one
thing, I have no idea how the integral closure in the field of
fractions condition is relevant.

Obviously one direction of the implication is so trivial it goes
without saying.
Warning: homework related.

Hints:

Suppose x is integral over R. Let K be the quotient field of R.

(1) Let p be the minimal polynomial of x over K. By definition, p is
monic and the coefficients of p are in K.

(2) Argue that the coefficients of p are integral over R [think about
symmetric polynomials].

(3) But R is integrally closed, so ...

quasi
.

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