Algebra and Complex Numbers

A high point of the chapter is the theorem that states that every nonconstant polynomial has a (possibly complex) root, even if we allow thecoefficients to be complex numbers. This result, known as the Fundamental Theorem of Algebra, is one of the deepest and most remarkable theorems in mathematics and was first proved by the brilliant German mathematician, Carl Friedrich Gauss, at the age of 20. Although the theorem does not tell us what the roots are, we will devise methods for actually finding roots in the coefficients of the polynomial are integers.