# Fundamental theorem of algebra

Fundamental theorem of algebra. Every non-constant polynomial with complex coefficients has at least one complex root.

We will not prove this theorem.

We can deduce that every degree $$n$$ polynomial has exactly $$n$$ complex roots by an inductive argument. A degree $$n$$ polynomial has at least one root by the fundamental theorem of algebra, and by polynomial division, we can write it as a product of a linear factor and a degree $$n-1$$ polynomial. By this degree $$n-1$$ polynomial has at least one root, so this can be factorised to a linear factor and a degree $$n-2$$ polynomial, and so on. By induction, a degree $$n$$ polynomial has $$n$$ complex roots.