I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state:
Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers prime to $p,n>0$ and the fraction $m/n$ is in lowest terms. Then $$ \text{ord}\left(\frac{m}{n} p^v \right)=v $$
However, it offers no example of this and I can't see how such a representation is unique. For example, if the integer was $13$, then $m=n=1$ and $p=13,v=1$. However, if the integer was $10$, we have $$ \frac{2}{1}5^1 $$ as a possible representation and we also have $$ \frac{5}{1}2^1 $$ The problem is even worse with non-integer rationals $$ \frac{15}{7}=\frac{3}{7}5^1=\frac{5}{7}3^1=\frac{15}{1}7^{-1} $$ as there is also no statement on $v$. Is it intended that $v$ be negative? Even if that is the case, then we still have $$ \frac{1}{10}=\frac{1}{2}5^{-1}=\frac{1}{5}2^{-1} $$ and the representation is not unique. I checked the errata and it makes no mention of error here. So what have I misinterpreted?
Here the prime $\,p\,$ is fixed, not variable. Indeed they write on p. 49 "so we let $p$ be some prime...".
If $\,p\nmid a,b\,$ then said unique rep of $\ t =\dfrac{a}b\ $ is $\ \dfrac{a}b\cdot p^{\large \color{#c00}0},\,$ so $\ v_p(t) = \color{#c00}0.$