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,

*
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since equality is transitive. However, this contradicts field Axiom 4 that

*
and
*
must be distinct elements
*




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If you are having trouble with math proofs a great book to learn from is How to Prove It by Daniel Velleman:


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*



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A really awesome book that I highly recommend on how to study math and be a math major is Laura Alcock"s, How to Study as a Mathematics Major: