Results 1 to 2 of 2

Math Help - Irreducible polynomial

  1. #1
    Junior Member
    Joined
    Sep 2009
    From
    Johannesburg, South Africa
    Posts
    71

    Irreducible polynomial

    I want to show that x^2+i is irreducible over \mathbb{Q}(i)

    My try;
    If there is an expansion it should be lineair. Thus there exist a x \in \mathbb{Q}(i) so that x^2+i=0
    Than x= \sqrt{-i} or x=- \sqrt{-i} \Rightarrow x \notin \mathbb{Q}(i)?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by bram kierkels View Post
    I want to show that x^2+i is irreducible over \mathbb{Q}(i)

    My try;
    If there is an expansion it should be lineair. Thus there exist a x \in \mathbb{Q}(i) so that x^2+i=0
    Than x= \sqrt{-i} or x=- \sqrt{-i} \Rightarrow x \notin \mathbb{Q}(i)?

    Suppose \sqrt{i}\in\mathbb{Q}(i)=\{a+bi\;;\;a,b\in\mathbb{  Q}\} \Longrightarrow \sqrt{i}=a+bi\Longrightarrow i=a^2+2abi-b^2 .

    Check now that from the last equation we get that i is a rational, and thus a real, number...( hint: distinguish the cases ab=\frac{1}{2}\,,\,\,ab\neq \frac{1}{2} )

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: April 7th 2010, 02:23 PM
  2. Irreducible Polynomial 2
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: June 8th 2009, 10:22 PM
  3. please please please help me~~ irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 26th 2009, 11:26 PM
  4. Irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 20th 2009, 01:22 AM
  5. irreducible polynomial
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 2nd 2008, 09:01 AM

Search Tags


/mathhelpforum @mathhelpforum