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How to prove rational number is a commutative field

  1. Sep 10, 2011 #1
    I was wondering how to prove rational number is a commutative field.

    Personally, I didn't think the word "commutative" is necessary, how about others?

    Do I simply prove it is commutative under multiplication?
  2. jcsd
  3. Sep 10, 2011 #2
    You'll need to check the field axioms. See http://en.wikipedia.org/wiki/Field_(mathematics)

    That is, you need to check

    • Associativity of addition and multiplication
    • Commutativity of addition and multiplication
    • Existence of the additive and multiplicative identities
    • Existence of the additive and multiplicative inverses for each nonzero element
    • Distributivity

    So;, which one is troubling you??
  4. Sep 10, 2011 #3
    Hey, micromass. How do I check field axioms?
    Do I say a and b belongs to rational numbers and go through the axiom?
  5. Sep 10, 2011 #4
    OK, let me do an example. Let me check commutativity of addition. Take two fraction a/b and c/d. We must prove that






    because addition and multiplications in the integers is commutative, we got that the two right-hand sides above are equal. Thus the left-hand-sides are also equal. Thus addition is commutative.

    Can you prove all the other ones?
  6. Sep 10, 2011 #5
    Thanks so much, I think I got it, that was very helpful! :)
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