Results 1 to 4 of 4

Math Help - distinct powers

  1. #1
    Junior Member
    Joined
    Feb 2010
    Posts
    34

    distinct powers

    Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by NikoBellic View Post
    Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
    Are you looking for this to be done group theoretically (as your language and notation seem to suggest) or number theoretically (as the section you posted in seems to suggest)?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by NikoBellic View Post
    Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
    My Pell's Equation argument in your other post would show there are infinite inverses, but that argument kind of sucks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by NikoBellic View Post
    Show that the powers of sqrt(2) + 1 are all distinct, and so (unlike Gaussian integers) there are infinitely many invertible elements in Z[sqrt(2)].
    Suppose  (1+\sqrt2)^n=(1+\sqrt2)^k where  n>k .

    We then would have  (1+\sqrt2)^{n-k}=1 which is impossible since  1+\sqrt2>1\implies (1+\sqrt2)^{n-k}>1


    Now observe  (1+\sqrt2)^n\cdot (\sqrt2-1)^n = 1 .
    Last edited by chiph588@; April 24th 2010 at 12:34 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: December 19th 2011, 09:34 AM
  2. How many distinct prime powers divide n ?
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: June 10th 2010, 10:12 PM
  3. Replies: 8
    Last Post: March 4th 2010, 11:09 AM
  4. three distinct
    Posted in the Algebra Forum
    Replies: 2
    Last Post: July 7th 2008, 11:35 PM
  5. Replies: 6
    Last Post: April 25th 2008, 08:23 AM

Search Tags


/mathhelpforum @mathhelpforum