Results 1 to 3 of 3

Math Help - field extension question

  1. #1
    Newbie pc31's Avatar
    Joined
    Jun 2008
    From
    Maine, US
    Posts
    12

    field extension question

    So i need to prove that the extension field:
    Q(sqrt(2)+sqrt(3)) = the set {a + b.sqrt(2) + c.sqrt(3) + d.sqrt(6) ; with a,b,c,d are rational numbers}

    Since Q(sqrt(2)+sqrt(3)) is the SMALLEST field that contains all rationals and sqrt(2)+sqrt(3) hence Q belongs to the set

    However, I dont know how to prove the other way around, i.e. the set belongs to Q(sqrt(2)+sqrt(3)).

    Can you please help me? Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Joined
    Sep 2008
    Posts
    34
    I wonder if you can use something like synthetic division -
    a + b rt 2 + c rt 3 + d rt 6 =
    (m + n rt 2)(p + q rt 3) + r + s rt 2 + t rt 3
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by pc31 View Post
    So i need to prove that the extension field:
    Q(sqrt(2)+sqrt(3)) = the set {a + b.sqrt(2) + c.sqrt(3) + d.sqrt(6) ; with a,b,c,d are rational numbers}
    I am not sure what you are trying to prove.
    I assume it is \mathbb{Q}(\sqrt{2},\sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3}).

    Trivially, \mathbb{Q}(\sqrt{2}+\sqrt{3}) \subseteq \mathbb{Q}(\sqrt{2},\sqrt{3})

    Now \frac{1}{\sqrt{2}+\sqrt{3}} = \sqrt{2} - \sqrt{3} \in \mathbb{Q}(\sqrt{2}+\sqrt{3}).
    Thus, \tfrac{1}{2}[(\sqrt{2}-\sqrt{3}) + (\sqrt{2}+\sqrt{3}) ]= \sqrt{2} \in \mathbb{Q}.
    Similarly, \sqrt{3}\in \mathbb{Q}.

    Thus. \mathbb{Q}(\sqrt{2},\sqrt{3})\subseteq \mathbb{Q}(\sqrt{2}+\sqrt{3})

    Putting this together we get \mathbb{Q}(\sqrt{2},\sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3}).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. a question on field extension
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 19th 2011, 04:07 PM
  2. Extension Field
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 12th 2011, 12:31 AM
  3. Field extension question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 2nd 2011, 02:37 AM
  4. Field extension question
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 18th 2009, 11:42 AM
  5. Field Extension
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 30th 2008, 02:52 PM

Search Tags


/mathhelpforum @mathhelpforum