Fundamental Theorem of Algebra

theoremabstract-algebra

Intuition

We know this from as early as pre-algebra! And note that this doesn’t hold for general cases in $\mathbb{R}$.

Formal Statement

Every non-constant polynomial with complex coefficients has a zero in $\mathbb{C}$.

Corollaries

If $p\in P(\mathbb{C})$ is a non-constant polynomial, then $p$ has a unique factorization (up to order) of the form,

$$p(z)=c(z-{\lambda }_1)\cdots (z-{\lambda }_m),$$

where $c,{\lambda }_1,\dots ,{\lambda }_m$ are complex numbers.