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)

  • 4
    $\begingroup$ Tame inertia is prime-to-$p$ by definition, and is isomorphic to $\prod_{\ell \neq p} \mathbb Z_{\ell}$. $\endgroup$
    – Emerton
    Mar 3 '11 at 8:33
  • $\begingroup$ 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 ? $\endgroup$ Mar 3 '11 at 11:55
  • 1
    $\begingroup$ The answer to your new first question is "infinite". An easy computable upper bound for your second question is $\log \ell/\log p$. Standard conjectures about the asymptotic behavior of primes suggest that this bound is sharp up to subtraction by a constant. I am thinking of closing this question. $\endgroup$
    – S. Carnahan
    Mar 3 '11 at 13:45

I think the tame ramification group is isomorphic to $\prod_{\ell\neq p}\mathbb{Z}_\ell$, not $\prod_{\ell\neq p}\mathbb{Z}_\ell^\times$. See for example, pp.12 of the book "Gauss sums, Kloosterman sums, and Monodromy Groups" by N. Katz.


  • 4
    $\begingroup$ The structure of the maximal tamely ramified extension of a local field was determined before N. Katz was born. See for example Hasse's Number Theory (Chapter 16) or jstor.org/stable/1992998 $\endgroup$ Mar 3 '11 at 12:05

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