Results 1 to 3 of 3

Math Help - Quotient Fields

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    58

    Quotient Fields

    The problem states:

    Determine Q(D) for D = {m + n√2 | m,n in Z} where D is an intefral domain and Q(D) its quotient field.

    I know that the answer is Q(D) Q(√2), but am not sure how to come to this conclusion.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by page929 View Post
    The problem states:

    Determine Q(D) for D = {m + n√2 | m,n in Z} where D is an intefral domain and Q(D) its quotient field.

    I know that the answer is Q(D) Q(√2), but am not sure how to come to this conclusion.

    Hint: assuming \displaystyle{a+b\sqrt{2}\neq 0\,,\,\,\frac{m+n\sqrt{2}}{a+b\sqrt{2}}=\frac{ma-2nb+(an-mb)\sqrt{2}}{a^2-2b^2}}

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45
    We can also use that Q(D) is the smallest field containing D (up to isomorphism) and \mathbb{Q}(\sqrt{2}) the smallest field containing \mathbb{Q}\cup \{\sqrt{2}\} (up to isomorphism) .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. vector fields - graphing gradient fields
    Posted in the Calculus Forum
    Replies: 0
    Last Post: March 20th 2010, 06:53 PM
  2. Quotient Rings and Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 18th 2010, 12:09 PM
  3. Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: September 23rd 2009, 07:12 PM
  4. Quotient Fields
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: May 13th 2009, 11:49 PM
  5. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 08:29 AM

Search Tags


/mathhelpforum @mathhelpforum