Re: Another algebra question
- From: Snis Pilbor <snispilbor@xxxxxxxxx>
- Date: Sun, 30 Sep 2007 09:47:29 -0700
On Sep 30, 9:10 am, quasi <qu...@xxxxxxxx> wrote:
On Sun, 30 Sep 2007 01:55:37 -0700, Snis Pilbor <snispil...@xxxxxxxxx>
wrote:
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.
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
Interesting, but this only shows that the minimal polynomial of x over
K are elements of R. Couldn't x have a different minimal polynomial
over F?
.
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