Question

Prove that field has no zero divisors

Answer 1

Answer to Question #82580 - Math / Abstract Algebra

Question. Prove that field has no zero divisors.

Answer. Let $K$ be a field and $a \in K$ be a zero divisor. By definition of zero divisor, there is $b \in K \setminus \{0\}$ such that $ab = 0$, and $a \neq 0$. By definition of field, every non-zero element of $K$ has an inverse, so there is $b^{-1}$. Multiplying $ab = 0$ by $b^{-1}$, we have

$$a = a b b^{-1} = 0 \cdot b^{-1} = 0,$$

contradiction. Therefore, there are no zero divisors in $K$.

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