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## Homework Statement

Suppose the field axioms include 0

^{-1}. Prove that, in this case, every element is equal to 0. Thus the existence of 0

^{-1}would contradict the field axiom that 1≠0.

## Homework Equations

## The Attempt at a Solution

My question regarding the proof is, why bother to show that

*every*element in the field is 0 in order to show that 1≠0. In other words, isn't it easier to say:

**Previously proven lemma:**For all x in F, 0*x = 0.

Suppose there exists 0

^{-1}in F such that 0*0

^{-1}=1. Then by the lemma above, the left side of the previous equation simply reduces to 0, and hence we are left with 1 = 0, a contradiction.