I'm trying to understand how a valuation $v$ of a field $K$ extends to an algebraic extension $L$. In chapter 8 of his book ANT, Neukirch first chooses a $K$-embedding $\tau : L \longrightarrow \bar{K}_v$ where $\bar{K}_v$ denotes an algebraic closure of the completion of $K$ wrt $v$. He then defines a $K$-embedding $w=\bar{v} \circ \tau$ which is an extension of the valuation to $L$ and then considers the completion $L_w$. $\tau$ is clearly continuous wrt this valuation.
If $L/K$ is infinite, the localisation $L_w$ is defined to be the union $\bigcup_iL_{iw}$ of the completions $L_{iw}$ of all the finite subextensions $L_i/K$ of $L/K$.
I have two questions:
1) How does $\tau$ extend uniquely to a continous $K$-embedding $\tau : L_w \longrightarrow \bar{K}_v$ in the case that $L/K$ is finite?
2) If $L/K$ is infinite, is the above union a field? Also how does $\tau$ extend uniquely in this case?
Any help would be much appreciated.