ring without irreducibles

An integral domain may not anyirreducible elements.  One such example is the ring of allalgebraic integers.  Any nonzero non-unit $\vartheta$ of this ring satisfies an equation

$$x^n+a_1{x}^{n-1}+\mathrm{\cdots }+a_{n-1}x+a_n=0$$

with integer coefficients $a_j$, since it is an algebraicinteger; moreover,  we can assume that $a_n=\text{N}(\vartheta )\ne \pm 1$  (see norm and trace ofalgebraic number: 2).  The element$\vartheta$ has the

$$\vartheta =\sqrt{\vartheta }\cdot \sqrt{\vartheta }.$$

Here, $\sqrt{\vartheta }$ belongs to the ring because itsatisfies the equation

$$x^{2n}+a_1{x}^{2n-2}+\mathrm{\cdots }+a_{n-1}{x}^2+a_n=0,$$

and it is no unit.  Thus the element $\vartheta$ is not irreducible.