ring without irreducibles

An integral domainMathworldPlanetmath may not any irreducible elementsMathworldPlanetmath.  One such example is the ring of all algebraic integersMathworldPlanetmath.  Any nonzero non-unit ϑ of this ring satisfies an equation


with integer coefficients aj, since it is an algebraic integer; moreover,  we can assume that  an=N(ϑ)±1  (see norm and trace of algebraic number: 2).  The element ϑ has the


Here, ϑ belongs to the ring because it satisfies the equation


and it is no unit.  Thus the element ϑ 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