I was studying ramification index and I found the following definition:
Let $E/K$ a finite field extension, $\mathfrak{d}$ a DVR of $K$ with maximal ideal $m$ and $\mathfrak{D}$ a DVR of $E$ with maximal ideal $M$ such that $\mathfrak{d}=\mathfrak{D}\cap K$ and $m=M\cap K$. If $v_\mathfrak{d}$ and $v_\mathfrak{D}$ are the valuation of $\mathfrak{d}$ and $\mathfrak{D}$ respectively then the ramification index is defined as $|v_\mathfrak{D}(E\setminus\lbrace 0\rbrace):v_\mathfrak{d}(K\setminus\lbrace 0\rbrace)|$.
Equvialently if $m=(t)$ and $M=(u)$, the ramification index is the positive integer $e$ such that $t=u^ev$, with $v$ a unit of $\mathfrak{D}$.
I understand the definition, but it seems something very hard to compute. The examples I have found on internet are some difficut ones using $p$-adic numbers. I would like if you can give me two elementary examples (avoiding $p$-adic numbers) one where there are no ramification (ramification index $=1$) and one with ramification (ramification index $>1$).
Thanks for your help.