Let $p$ and $\ell$ be distinct rational primes. Note that the unit group of the finite field $\mathbb{F}_\ell$

is of order $\ell-1$, hence there is the probability of finding a $p$-quotient from $\mathbb{F}_\ell^\times$.

When taking the infinite product $\prod_{\ell\neq p}\mathbb{F}_\ell^\times$, how large could its maximal pro-p quotient be? And do we have a comuptable bound for the p-valuation of $\ell -1$ when $\ell$ tends to infinity?

(My original question was about pro-p quotient of the tame inertia group of a p-adic local field, which didn't make sense for obvious reason (pointed out as below), and I propose the replace it with this one)

And do we have a comuptable bound...As there is an $l$ which is $\equiv1\pmod{p^n}$ for every $n>0$, how can there be such a bound ? $\endgroup$