Galois Theory, 5th Edition

Chapter 2The Fundamental Theorem of Algebra

DOI: 10.1201/9781003213949-2

At the time of Galois, the natural setting for most mathematical investigations was the complex number system. The real numbers were inadequate for many questions, because $-1$ has no real square root. The arithmetic, algebra, and—decisively—analysis of complex numbers were richer, more elegant, and more complete than the corresponding theories for real numbers.

In this chapter we establish one of the key properties of $ℂ$, known as the Fundamental Theorem of Algebra. This theorem asserts that every polynomial equation with coefficients in $ℂ$ has a solution in $ℂ$. This theorem is, of course, false over $ℝ$ —consider the equation $t^2+1=0$. It was fundamental to classical algebra, ...