ring without irreducibles

An integral domain may not any irreducible elements.  One such example is the ring of all algebraic 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 algebraic integer; moreover,  we can assume that  $a_n=\text{N}(\vartheta )\ne \pm 1$  (see norm and trace of algebraic number: 2).  The element $\vartheta$ has the

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

Here, $\sqrt{\vartheta }$ belongs to the ring because it satisfies 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.

Title	ring without irreducibles
Canonical name	RingWithoutIrreducibles
Date of creation	2014-05-29 11:39:19
Last modified on	2014-05-29 11:39:19
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	15
Author	pahio (2872)
Entry type	Example
Classification	msc 13G05
Related topic	FieldOfAlgebraicNumbers