Let $(R,M)$ be DVR and $v$ be it's valuation. And $p$ be a character of its residue field($p$＝$0$ or prime).
In the case $p≠0$(p is positive prime),can we say $v(p)＝∞$ ?
If residue fields's character is equal to $p$,$v(p)＝v(0)＝∞$ is obvious. But what about the residue field's character is not equal to $p$ ? (mixed character case)
Thank you in advance.
Part of the definition of a valuation is that $v(a) = \infty$ if and only if $a = 0$. So if $v(p) = \infty$ then we would necessarily have $p = 0$ in $R$. That would require $R$ to have characteristic $p$ (or $1$ but that's a degenerate case).
For a concrete example, take $R = \mathbb Z_p$ the $p$-adic integers and $v$ the corresponding $p$-adic valuation. The residue field of $\mathbb Z_p$ is $\mathbb Z/p \mathbb Z$, which of course has characteristic $p$. However, $v(p) = 1 \neq \infty$. In fact, $\mathbb Z_p$ has characteristic $0$.