# irreducible polynomials

• April 25th 2008, 11:51 AM
JCIR
irreducible polynomials
For polynomials over a field F, prove that every non constant polynomials can be expressed as a product of irreducible polynomials.
• April 25th 2008, 12:35 PM
ThePerfectHacker
Quote:

Originally Posted by JCIR
For polynomials over a field F, prove that every non constant polynomials can be expressed as a product of irreducible polynomials.

If $f(x)$ is irreducible the proof is complete. Otherwise $f(x) = p_1(x)p_2(x)$ where $p_1(x),p_2(x)$ are not constant. If these two polynomials are irreducible the proof is complete. Otherwise keep on going. Eventually you end up with only irreducible polynomials. Because the sums of the degrees need to add up to $\deg f(x)$ and this means you cannot do this indefinitely.