Consider $\Bbb Q_3$ and its quadratic extension $\Bbb Q_3(\sqrt{3})$. I want to find the ramification index $e$.
We have the valuation groups as follows:
For $\Bbb Q_3$, the valuation group $G=\{|a|_3: a \in \Bbb Q_3, \ a \neq 0 \}$ is generated by $3$,
For $\Bbb Q_3(\sqrt{3})$, the valuation group $G'=\{|b|: b \in \Bbb Q_3, \ b \neq 0 \}$ is generated by $\sqrt 3$.
Thus, $e=\text{number of elements of $G'/G$}= ?$
As $\Bbb Q_3(\sqrt 3)$ is ramified, so $e=2$.
But how $e=\text{number of elements of $G'/G$}=2$ ?
Can you explain it please?