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Homework Help: Sum of fields is never a field

  1. Mar 30, 2010 #1
    1. The problem statement, all variables and given/known data
    Prove that a direct sum of two or more field is never a field

    2. Relevant equations

    F X G = {(f,g): f in F, g in G}

    3. The attempt at a solution

    I know that I need to prove FXG is Abelian group under addition, and FXG - {0,0} is an Abelian group under mult.
    And for mult,
    I know I need to check 1) mult identity, 2) closure, 3) commutative, 4) mult inverse, 5) associativity

    I have problem proving mult inverse,

    let a1, a2, b1, b2 be elements in F XG - {0,0}, such that (a1,b1)*(a2,b2) = (a1a2, b1b2)
    so (a1a2, b1b2) ^-1 = (1/(a1a2), 1/(b1b2)) which is not in FXG - (0,0)

    Is this right?
    Last edited: Mar 30, 2010
  2. jcsd
  3. Mar 30, 2010 #2
    How do you know a1a2 is invertible?
  4. Mar 30, 2010 #3
    Remember you're trying to prove that [itex]F\times G[/itex] is not a field.

    So you're not actually trying to prove any of the things you say you're trying to prove. You're trying to disprove at least one of them. So if you've had a problem proving one of them, that might give you a clue.
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