Question

3.4. Prove that any field is an integral domain.

Answer 1

Let $a\ne0$ and $b$ be two elements in the field $F$ and $ab=0$.

Since $F$ is a field and $a\ne0$ we have $a^{-1}\in F$. Hence $a^{-1}ab=a^{-1}0=0$.

So we obtain $b=0$.

Hence there exists no zero divisor in $F$.