Irreducible Polynomial - Simple Examples

Simple Examples

The following five polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

,

,

,

,

.

Over the ring of integers, the first two polynomials are reducible, the last two are irreducible. (The third, of course, is not a polynomial over the integers.)

Over the field of rational numbers, the first three polynomials are reducible, but the other two polynomials are irreducible.

Over the field of real numbers, the first four polynomials are reducible, but is still irreducible.

Over the field of complex numbers, all five polynomials are reducible. In fact, every nonzero polynomial over can be factored as

where is the degree, the leading coefficient and the zeros of . Thus, the only non-constant irreducible polynomials over are linear polynomials. This is the Fundamental theorem of algebra.

The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers ( ), to the real subset of the algebraic numbers ( ), and finally to the algebraic subset of the complex numbers ( ). After the invention of calculus those latter two subsets were later extended to all real numbers ( ) and all complex numbers ( ).

For algebraic purposes, the extension from rational numbers to real numbers is too "radical": it introduces transcendental numbers, which are not the solutions of algebraic equations with rational coefficients. These numbers are not needed for the algebraic purpose of factorizing polynomials (but they are necessary for the use of real numbers in analysis). The set of algebraic numbers ( ) is the algebraic closure of the rationals, and contains the roots of all polynomials (including i for instance). This is a countable field and is strictly contained in the complex numbers – the difference being that this field ( ) is "algebraically complete" (as are the complex numbers, ) but not analytically complete since it lacks the aforementioned transcendentals.

The above paragraph generalizes in that there is a purely algebraic process to extend a given field F with a given polynomial to a larger field where this polynomial can be reduced into linear factors. The study of such extensions is the starting point of Galois theory.