Algebra: A Very Short Introduction by Peter M. Higgins

Author:Peter M. Higgins
Language: eng
Format: azw3
ISBN: 9780198732822
Publisher: Oxford University Press
Published: 2015-03-15T18:38:04.437808+00:00

The Fundamental Theorem of Algebra asserts that any non-constant polynomial p(x) has a root, λ, which may be real or complex. The theorem is true quite generally—the coefficients of p(x) may be real or complex numbers. The theorem has a long history and is difficult to prove rigorously; no proof is completely algebraic, but rather there is always some element of a spatial, or, as it is called in mathematics, topological, argument involved. Although not a purely algebraic result, the Fundamental Theorem of Algebra has important consequences for the nature and associated factorizations of polynomials, as we now explain.

Let p(x) be a typical nth-degree polynomial, and suppose z is a complex root of p(x) so that a0 + a1z + … + anzn = 0. Take the conjugate of both sides of this equation. Of course, and, applying the rules that the conjugate of a sum is the sum of the conjugates and the conjugate of a product is the product of the conjugates, we obtain

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