Integral Extensions, Units in an Integral Extension
Units in an Integral Extension
Let S be an integral extension of R,
and let x be an element of R.
Now x is a unit in R iff it is a unit in S.
One direction is obvious, so assume x is a unit in S, with xy = 1.
Since y is integral over R,
p(y) = 0 for some monic polynomial p.
Multiply p(y) through by xn-1, and y lies in R after all.
Thus x is a unit in R.