Results 1 to 2 of 2

Math Help - Two difficult proofs (Fields)

  1. #1
    Junior Member
    Joined
    Jun 2009
    Posts
    58

    Exclamation Two difficult proofs (Fields)

    Help pleaseĦĦĦ

    1.-Prove that each subfield of the field of complex numbers contains every
    rational number.

    2.-Prove that each field of characteristic zero contains a copy of the rational
    number field.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    ynj
    ynj is offline
    Senior Member
    Joined
    Jul 2009
    Posts
    254
    1: Let Fbe a field. 0,1\in F\Rightarrow N\subseteq F\Rightarrow -N\subseteq F\Rightarrow Z\subseteq F\Rightarrow \frac{1}{Z^*}\subseteq F\Rightarrow \frac{p}{q}\in F\Rightarrow Q\subseteq F
    2:let Fbe a field, ebe its identity. Let F'=\{x|x\in F,x=pe(qe)^{-1},p,q\in Z,q!=0\},\phi:F'\rightarrow Qto be \phi(ne)=n, \phi(pe(qe)^{-1})=\frac{p}{q}(q!=0,n,p,q\in Z)
    ne are distinct from each other since if ne=me then (n-m)e=0\Rightarrow (n-m)a=0\forall a\in F, which will contradict to char(F)=0
    it is easy to show that F'is a subfield of F and \phiis a 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. Fields
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 25th 2009, 07:07 PM
  3. Fields
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 18th 2009, 05:46 PM
  4. Extension fields / splitting fields proof...
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: December 19th 2007, 08:29 AM
  5. Replies: 3
    Last Post: October 6th 2007, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum