Results 1 to 3 of 3

Math Help - Integral domain

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    14

    Integral domain

    Quote Originally Posted by The problem
    Suppose D is an integral domain with the property that there is a map \setminus \{0\} \to \mathbb{N}" alt="v\setminus \{0\} \to \mathbb{N}" /> satisfying the properties (i) For all a,b in D\0, v(a)v(b) = v(ab); and (ii) v(a) = 1 if and only if a is a unit of D. Show that D[x] also has a map with this property.
    I've tried all sorts of things, such as (call my map on D[x]\0 V)
    V(a_0 + a_1x + a_2x^2 + \ldots + a_nx^n) = 2^{n}(v(a_0) + v(a_1) + \ldots + v(a_n))
    (the 2^n is to distinguish a_0 being a unit from a_1, say, being a unit for property (ii)), but I'm always hitting a snag for property (i) since there isn't linearity of v. I know the integral domain \mathbb{Z}[\sqrt{-5}] is an example of such a D, with v(a+b\sqrt{-5}) = a^2 + 5b^2 so I've been playing with (\mathbb{Z}[\sqrt{-5}])[x] to help come up with a desirable V, but to no avail.

    If you could suggest what the map should like I'd be most appreciative.

    Many thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by james123 View Post
    I've tried all sorts of things, such as (call my map on D[x]\0 V)
    V(a_0 + a_1x + a_2x^2 + \ldots + a_nx^n) = 2^{n}(v(a_0) + v(a_1) + \ldots + v(a_n))
    (the 2^n is to distinguish a_0 being a unit from a_1, say, being a unit for property (ii)), but I'm always hitting a snag for property (i) since there isn't linearity of v. I know the integral domain \mathbb{Z}[\sqrt{-5}] is an example of such a D, with v(a+b\sqrt{-5}) = a^2 + 5b^2 so I've been playing with (\mathbb{Z}[\sqrt{-5}])[x] to help come up with a desirable V, but to no avail.

    If you could suggest what the map should like I'd be most appreciative.

    Many thanks

    An idea: if f(x)=a_0 + a_1x + a_2x^2 + \ldots + a_nx^n\,,\,\,a_n\neq 0 is a pol. in D[x] of degree n, try V(f(x)):= v(a_n) .

    Tonio

    Disclaimer: I'm not 100% sure the above works...not even 93.156% sure, in fact.


    Hmmm...now I'm almost sure it doesn't work because of condition (2)....
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    14
    Slapping on 2^n (ie. 2^{\deg(f)}) at the front fixes that (which I suspect you knew ). Thanks, Tonio!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: May 22nd 2011, 12:23 PM
  2. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 12th 2010, 01:47 AM
  3. Integral Domain
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 15th 2009, 08:33 PM
  4. integral domain?
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 23rd 2008, 10:03 AM
  5. In an integral domain
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 11th 2007, 08:18 PM

Search Tags


/mathhelpforum @mathhelpforum