For $x\in L$, choose a monic equation over $K$,
$$
x^n+c_1x^{n-1}+\cdots+c_n=0.
$$
Choose nonzero $a\in R$ with $ac_i\in R$ for every $i$. Multiplying by $a^n$ shows that $b=ax$ satisfies
$$
b^n+(ac_1)b^{n-1}+(a^2c_2)b^{n-2}+\cdots+a^nc_n=0,
$$
whose coefficients lie in $R$. Thus $b\in\overline R$ and
$$
\boxed{x=b/a,\qquad b\in\overline R,\ a\in R.}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 101 / 2 / a / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-101/2/a/iv/solution