Results 1 to 3 of 3

Math Help - Field

  1. #1
    Newbie
    Joined
    Feb 2010
    From
    Bangladesh
    Posts
    10

    Field

    I need the proof:

    The set of Rational numbers has no proper sub-field.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Apr 2010
    Posts
    78
    Q is a prime field of characteristic 0, i.e. Q can be embedded in any field of characteristic 0.
    But subfield preserves characteristic.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member nimon's Avatar
    Joined
    Sep 2009
    From
    Edinburgh, UK
    Posts
    64
    Alternatively, if F\subseteq\mathbb{Q} is a subfield then 1\in F, so that 1+1,1+1+1,\ldots\in F so \mathbb{N}\subseteq F. But then -1,-1-1,\ldots \in F so that \mathbb{Z}\subseteq F. But for each 0\neq z\in\mathbb{Z} we have that z^{-1} \in F, so that pq^{-1} \in F for any p,q\in\mathbb{Z},q\neq 0. Hence \mathbb{Q}\subseteq F.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring, field, Galois-Field, Vector Space
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 15th 2012, 04:25 PM
  2. Splitting Field of a Polynomial over a Finite Field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 1st 2011, 04:45 PM
  3. Z is not a field.
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: February 2nd 2011, 11:25 AM
  4. Field of char p>0 & splitting field
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 22nd 2009, 01:20 AM
  5. Field
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 12th 2006, 10:58 AM

Search Tags


/mathhelpforum @mathhelpforum