I have been reading this paper by Artin and Whaples on the Axiomatic Characterisation of Fields by the Product Formula for Valuations (https://projecteuclid.org/euclid.bams/1183507128) and on pg $475$ (seventh page of the linked PDF document), while defining the normed valuation of a field element corresponding to a prime divisor (a set of equivalent non-trivial valuations on the field), they say "where $v$ is the ordinal number of $\alpha$ at $\mathfrak{p}$". I am not sure what the rigorous definition of "ordinal number" is in this context and a short google search is not giving me the meaning I am looking for.
From the computation done on pages $477-478$ (which has led to the deduction of the formula $|| \phi(z) ||_p = q^{-vf}$ it seems that this "ordinal number" for $\alpha=\phi(z)$ in the field $k_1(z)$ is the exponent of $p(z)$ in the (unique) factorization of $\phi(z)$ as $p(z)^v\psi(z)$ where $\psi(z) \in k_1(z)$ has neither numerator nor denominator divisible by $p(z)$, but I can't find out what the definition is in general.
Thanks for the help.