Prove from the field axioms that the additive identity, 0, has no multiplicative inverse.

* Proof. * The proof is by contradiction. Suppose otherwise, that there is some

such that

(this is the definition of multiplicative inverse). Then, by part (b) of

this exercise, we know that

for any

. Hence,

since equality is transitive. However, this contradicts field Axiom 4 that and must be distinct elements

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