Matrix Analysis

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) = t2 – 2t + 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.