Complex Roots of Polynomial with Real Coefficients occur in Conjugate Pairs

Theorem
Let $f \left({z}\right) = a_n z^n + a_{n - 1} z^{n - 1} + \cdots + a_1 z + a_0$ be a polynomial over complex numbers where $a_0, \ldots, a_n$ are real numbers.

Let $\alpha \in \C$ be a root of $f$.

Then $\overline \alpha$ is also a root of $f$, where $\overline \alpha$ denotes the complex conjugate of $\alpha$.

That is, all complex roots of $f$ appear as conjugate pairs.