Results 1 to 3 of 3

Math Help - P-adic valuation

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    170
    Thanks
    8

    P-adic valuation

    Hello,

    I've been reading about p-adics and I think I understand most of the basics (Thanks to David A. Madore's "A First Introduction to p-adic Numbers" found at http://www.madore.org/~david/math/padics.pdf).

    I need some clarification (maybe an example would help) on how to get the inverse of a p-adic integer.

    The method used (in the paper) is for getting the inverse of 1-\alpha. So, if you have a p-adic integer, say \gamma, which ends in 1, you have to let \gamma = 1 - \alpha, and solve for \alpha, which will end with a 0. Then the inverse of \gamma = 1 + \alpha + \alpha^2 + .... Is this the correct way of getting the inverse of \gamma?

    Now, if \gamma ends in something else other than zero or one, what we are supposed to do is look for a digit f so that f\gamma will end in a one, and thus can be inverted by the above method (if it's correct ). Since we have the inverse of f\gamma, to get the inverse of \gamma, we multiply the inverse of f\gamma by f again. Is this correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    If  \alpha ends in  d , pick the unique  f such that  0<f<p and  df\equiv1\bmod{p} .

    Now  \beta=f\alpha has a last digit of  1 . We then can find the inverse of  \beta .

    So  \beta\beta^{-1}=1\implies(\alpha)\cdot\left(f\beta^{-1}\right)=1\implies (\alpha)\cdot\left(f(f\alpha)^{-1}\right)=1\implies\alpha^{-1}=f(f\alpha)^{-1} .
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2010
    From
    Israel
    Posts
    148
    Thanks for the link!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. p-adic valuation question
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: January 8th 2012, 04:10 PM
  2. [SOLVED] p-adic metric
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 4th 2010, 05:29 AM
  3. P-adic Number help please
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: May 10th 2010, 08:04 AM
  4. P-adic numbers
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: April 15th 2010, 05:13 AM
  5. valuation ring, discrete valuation
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 18th 2008, 11:47 PM

Search Tags


/mathhelpforum @mathhelpforum