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Zero has no reciprocal

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 a \in \mathbb{R} such that a \cdot 0 = 1 (this is the definition of multiplicative inverse). Then, by part (b) of this exercise, we know that a \cdot 0 = 0 for any a. Hence,

    \[ a \cdot 0 = 1 \quad \text{and} \quad a \cdot 0 = 0 \qquad \implies \qquad 1 = 0 \]

since equality is transitive. However, this contradicts field Axiom 4 that 0 and 1 must be distinct elements. \qquad \blacksquare

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