The fundamental theorem of algebra

Abstract

One historical motivation for introducing the complex numbers C was that polynomials with real coefficients might not have real zeroes. For example, a calculation reveals that {1 + i , 1 − i } are zeroes of the polynomial p ( t ) = t 2 – 2 t + 2, which has no real zeroes. All zeroes of any polynomial with real coefficients are, however, contained in C . In fact, all zeroes of all polynomials with complex coefficients are in C . Thus, C is an algebraically closed field : There is no field F such that C is a subfield of F , and such that there is a polynomial with coefficients from C and with a zero in F that is not in C . The fundamental theorem of algebra states that any polynomial p with complex coefficients and of degree at least 1 has at least one zero in C . Using synthetic division, if p ( z ) = 0, then t − z divides p ( t ); that is, p ( t ) = ( t − z ) q ( t ), in which q ( t ) is a polynomial with complex coefficients, whose degree is 1 smaller than that of p . The zeroes of p are z , together with the zeroes of q . The following theorem is a consequence of the fundamental theorem of algebra. Theorem . A polynomial of degree n ≥ 1 with complex coefficients has, counting multiplicities, exactly n zeroes among the complex numbers . The multiplicity of a zero z of a polynomial p is the largest integer k for which ( t − z ) k divides p ( t ). If a zero z has multiplicity k , then it is counted k times toward the number n of zeroes of p . It follows that a polynomial with complex coefficients may always be factored into a product of linear factors over the complex numbers.