Re: p-adic transcendentals

From: G. A. Edgar (edgar_at_math.ohio-state.edu)
Date: 10/22/04 

Date: 22 Oct 2004 09:33:51 -0400

> David Madore a ecrit:
> >
> > So, is there a known way of proving that log(3)/log(2) is
> > transcendental in Q_p?
> >

Is there a sense in which this p-adic number x satisfies 2^x = 3 ?

serge bouc <sergepointbouc@free.fr> wrote:
> I don't know the answer, but your question reminds me the following,
> [roughly translated from Exercice 4 of Chapter IV Û5 of
> Borevitch-Chafarevitch "Theorie des Nombres"] :
>
> "Find a mistake in the following proof of the irrationality of pi.
> The number pi is the smallest number >0 such that sin(pi)=0.
> Suppose that pi is rational. Then since pi>3, the numerator of pi
> has to be divisible by some odd prime p, or by 2^2. In that case
> set p=2. It follows that the series sin(x) and cos(x) are convergent
> in Q_p at x=pi. but since
> sin(x+y)=sin(x)cos(y)+sin(y)cos(x)
> it follows that sin(n pi)=0 for any integer n. Hence the function
> sin(x) has infinitely many zeros inside its convergence area,
> hence it is identically zero (from Exercice 1 at the same page).
> This is a contradiction".
>

I guess the error is: A series of rationals that convervges both in
the usual absolute value and in a p-adic absolute value need not
converge to the same thing in the two cases, even if both limits
are rationals.

Relevant Pages

  • Re: p-adic transcendentals
    ... >Write a sequence of rationals x_n that converges to 0 in the usual absolute ... r_p for all primes p, you can find a sequence of rationals ...
    (sci.math)
  • Re: p-adic transcendentals
    ... > I guess the error is: A series of rationals that convervges both in ... > the usual absolute value and in a p-adic absolute value need not ... > are rationals. ... concerning this exercise. ...
    (sci.math.research)
  • Re: Cantor and the binary tree
    ... >> It is also sufficient to consider all rationals as has been done! ... number than any convergent infinite sequence. ... anti-diagonal as well are rational fractions so they exist as much as ...
    (sci.math)
  • Re: p-adic transcendentals
    ... Write a sequence of rationals x_n that converges to 0 in the usual absolute ... Can you specify in advance lots of rational limits, ...
    (sci.math)