Question related to trace

Question related to trace

pI was reading this document called Discriminants and Ramified Primes: a
href=http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/disc.pdf
rel=nofollowhttp://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/disc.pdf/a/p
pSuppose $O_K$ is the ring of algebraic integers of a number field $K$.
Let $\mathcal{P}$ be a prime ideal of $O_K$ above $p$ such that
$O_K/\mathcal{P}$ is a finite field of characteristic $p$. Near the bottom
of page 4 it states: We want to prove
$\text{disc}_{\mathbb{Z}/p\mathbb{Z}} (O_K/\mathcal{P}) \not = \bar{0}$.
If this discriminant is $\bar{0}$ then (because $O_K/\mathcal{P}$ is a
field) the trace function $\operatorname{Tr} : O_K / \mathcal{P}
\rightarrow \mathbb{Z}/p\mathbb{Z}$ is identically zero. /p pI don't quite
understand why the trace would be identically zero in this situation.
Could someone give me an explanation for this? Thank you very much!/p