extension of a valuation

《Algebraic Number Theory》 · 第 161–162 页 · Chapter II，Section 8「Extensions of Valuations」，Extension Theorem 8.1

Let $L/K$ be an algebraic field extension and $v$ a valuation of $K$. Then: (i) Every extension $w$ of the valuation $v$ arises as the composite $w=v\circ\tau$ for some $K$-embedding $\tau:L\to\overline K_v$. (ii) Two extensions $v\circ\tau$ and $v\circ\tau'$ are equal if and only if $\tau$ and $\tau'$ are conjugate over $K_v$.