Integral closure in a finite extension of a complete discretely valued field
= Integral closure in a finite extension of a complete discretely valued field
Let $K$ be complete for a discrete valuation and let $L/K$ be finite. The <valuation ring> $\mathcal O_L$ is exactly the integral closure of $\mathcal O_K$ in $L$. One direction follows from a monic integral equation; for the other, the coefficients of the <minimal polynomial> are elementary <symmetric polynomials> in conjugates of absolute value at most one.