What Are Algebraic Numbers and Minimal Polynomials?

An algebraic number is a root of a polynomial with rational coefficients. For example, $\sqrt{2}$ and $i = \sqrt{-1}$ are algebraic numbers, because they are roots of the quadratics $x^2 - 2$ and $x^2 + 1$, respectively.

Given an algebraic number $z$, a minimal polynomial $m$ of $z$ is a polynomial with rational coefficients, of the lowest degree possible, such that $m(z) = 0$. For example, even though $i$ is a root of $x^3 + x$, the polynomial $x^3 + x = x(x^2 + 1)$ is not a minimal polynomial of $i$, because $i$ is a root of a smaller degree polynomial $x^2 + 1$. However, $x^2 + 1$ is a minimal polynomial of $i$.