Theorem 5.2. Assuming that:

$(K, | ∙ |)$ is a complete discretely valued field
such that $k : = O_{K} ∕ m$ is a perfect field of characterist $p$

Then there exists a unique map

[∙] : k \to O_{K}

such that

(i) $a \equiv [a] mod m$ for all $a \in k$
(ii) $[a b] = [a] [b]$ for all $a, b \in k$

Moreover if $characteristic O_{K} = p$ , then $[∙]$ is a ring homomorphism.