# Homomorphisms between fields are injective.

How would I prove this?

I know that I must show $$f(a)=f(b) \Rightarrow a = b$$

I also know I must use the definition of homomorphism, ie:

$$f(a+b)=f(a)+f(b)$$

$$f(ab)=f(a)f(b)$$

$$f(1)=1$$

I am assuming that a contradiction would be a good approach to solve this, but not quite sure on specifics.

• Duplicate. See here – Troy Woo Oct 4 '14 at 13:13
• I don't believe that this is a duplicate (or more accurately, I don't believe that the OP, given the way this question was asked, could possibly regard it as a duplicate). – John Hughes Oct 4 '14 at 13:26

Suppose $$f(a) = f(b)$$, then $$f(a-b) = 0 = f(0)$$. If $$u = (a-b) \ne 0$$, then $$f(u)f(u^{-1}) = f(1) = 1$$, but that means that $$0 f(u^{-1}) = 1$$, which is impossible. Hence $$a - b = 0$$ and $$a = b$$.
• I've defined $u = a-b$, and just established that $f(a-b) = 0$. So $f(u) = 0$. As for why $f(0) = 0$, it's because $f(0+1) = f(0) + f(1)$, which gives $f(1) = f(0) + f(1)$, so $1 = f(0) + 1$, so $0 = f(0)$. – John Hughes Oct 4 '14 at 13:23