Sum of fields is never a field

[rainiscoming]

Homework Statement

Prove that a direct sum of two or more field is never a field

Homework Equations

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

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?

 

[eok20]

How do you know a1a2 is invertible?

 

[Martin Rattigan]

Remember you're trying to prove that $F\times G$ 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.