Fundamental Theorem of Algebra

Fundamental Theorem of Algebra

This foundational (unproved, stated-as-fact) theorem comes in two equivalent forms:

• "A polynomial equation with real coefficients has at least one root" in $\mathbb{C}$.
• "A polynomial equation with complex coefficients and of degree $n$ has $n$ complex roots."

This is the deep reason a quadratic equation is never left without a solution, even when its discriminant is negative: the complex numbers are exactly large enough to guarantee every polynomial equation — of any degree, with real or even complex coefficients — has as many roots (counted with multiplicit …