Results 1 to 2 of 2

Math Help - Need HW help about norm and prime numbers

  1. #1
    Junior Member
    Joined
    Apr 2008
    From
    Gainesville
    Posts
    68

    Need HW help about norm and prime numbers

    Let p be either a prime number, or -1. Prove that the following norm defined on Z[sqrt(p)] (a + bsqrt(p), a,b are in Z )

    N(a+bsqrt(p))= /a^2 -pb^2/. ( the bars mean absolute value)

    If N(a+bsqrt(p)) is a prime number, then a+bsqrt(p) is irreducible.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by JCIR View Post
    Let p be either a prime number, or -1. Prove that the following norm defined on Z[sqrt(p)] (a + bsqrt(p), a,b are in Z )

    N(a+bsqrt(p))= /a^2 -pb^2/. ( the bars mean absolute value)

    If N(a+bsqrt(p)) is a prime number, then a+bsqrt(p) is irreducible.
    This norm has the following property: N(\alpha \beta) = N(\alpha)N(\beta), you can prove this directly by letting \alpha = a+b\sqrt{p} and \beta = c+d\sqrt{p} and multipling everything out.

    Let N(\alpha) is a prime number. Suppose that \alpha = \beta \gamma where neither \beta,\gamma are units. Then it would mean N(\alpha) =N(\beta)N(\gamma) with neither N(\beta),N(\gamma) being 1. Thus, we have factored N(\alpha) non-trivially, and so it cannot be a prime number. A contradiction. Thus, \alpha is irreducible.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 22nd 2011, 12:37 PM
  2. Prime numbers.
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: September 18th 2010, 04:42 PM
  3. Prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: April 26th 2010, 06:35 PM
  4. prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: February 14th 2007, 07:21 PM
  5. Prime Numbers
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: February 9th 2007, 12:53 PM

Search Tags


/mathhelpforum @mathhelpforum