Unique extension of an absolute value to a finite extension

If $K$ is complete and $L/K$ is finite, an absolute value on $K$ extends uniquely to $L$. With the normalization that it restricts exactly to the given absolute value,Every $K$-automorphism of $L$ preserves this extended absolute value.

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.