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

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- Apr 25th 2008, 11:51 AMJCIRirreducible polynomials
For polynomials over a field F, prove that every non constant polynomials can be expressed as a product of irreducible polynomials.

- Apr 25th 2008, 12:35 PMThePerfectHacker
If $\displaystyle f(x)$ is irreducible the proof is complete. Otherwise $\displaystyle f(x) = p_1(x)p_2(x)$ where $\displaystyle p_1(x),p_2(x)$ 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 $\displaystyle \deg f(x)$ and this means you cannot do this indefinitely.