Fundamental theorem of algebra

Not to be confused with Fundamental theorem of arithmetic or Fundamental theorem of linear algebra.

The fundamental theorem of algebra, also called d'Alembert's theorem[1] or the d'Alembert–Gauss theorem,[2] states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.

Despite its name, it is not fundamental for modern algebra; it was named when algebra was synonymous with the theory of equations.

  1. ^ Dunham, William (September 1991), "Euler and the fundamental theorem of algebra" (PDF), The College Journal of Mathematics, 22 (4): 282–293, JSTOR 2686228
  2. ^ Campesato, Jean-Baptiste (November 4, 2020), "14 - Zeroes of analytic functions" (PDF), MAT334H1-F – LEC0101, Complex Variables, University of Toronto, retrieved 2024-09-05