Exercise 1.39

Proof. Let $R$ an integral domain, and $\pi$ a prime in $R$.

If $\pi =\mathit{\alpha \beta },\alpha ,\beta \in R$, a fortiori $\pi$ divides $\mathit{\alpha \beta }$. As $\pi$ is a prime, $\pi$ divides $\alpha$ or $\beta$, say $\alpha$, so there exists $\xi \in R$ such that $\alpha =\mathit{\xi \pi }$, so $\pi =\mathit{\xi \pi \beta },\pi (1-\mathit{\xi \beta })=0$. As $A$ is an integral domain, and $\pi \ne 0$ by definition, $1=\mathit{\xi \beta }$, so $\beta$ is an unit. If $\pi =\mathit{\alpha \beta }$, $\alpha$ or $\beta$ is a unit, so $\pi$ is irreducible. □