Prime Factorization of Polynomials

Theorem

Let $f\in \mathbb{F}[x]$ be a non-scalar polynomial, Then, $f=\alpha \cdot p_1^{n_1}\cdot \cdots \cdot p_k^{n_k}$ where:

• $p_1,\dots ,p_k\in \mathbb{F}[x]$ are unique distinct Prime Polynomials
• $n_1,\dots ,n_k$ are positive integers
• $\alpha$ is a scalar

Example Prime Decomposition

Real Function

1. For $f(x)=x^3-1$
2. Then, $f(x)=(x-1)(x^2+x+1)$, and all of these factors are Prime Polynomials

Complex function

1. For $f(x)=x^3-1$
2. $x_1=\frac{1}{2}+\frac{\sqrt{3}}{2}i$
3. $x_2=\frac{1}{2}-\frac{\sqrt{3}}{2}i$
4. $f(x)=(x-1)(x-(-\frac{1}{2}+\frac{\sqrt{3}}{2}i))(x-(\frac{1}{2}-\frac{\sqrt{3}}{2}i))$