The Fundamental Theorem of Algebra via the Fourier Inversion Formula

The Fundamental Theorem of Algebra, or “FTA,” has a rich, centuries-old history [6]. The numerous proofs that can be found demonstrate the importance and continuing interest. Let the number of proofs be N (we can verify N > 80). We provide proofs N + 1 through N + 4 using the Fourier transform and its inversion formula, or “FIF.” A few reasons for so many proofs are: to have a proof by the most elementary means, a proof suitable for a class in math, and a proof by subject area. The FTA is a wellknown consequence of Liouville’s theorem, see [1, p. 122], a topic in complex variable math courses. This note is completed by showing FIF implies Liouville’s theorem. Our proofs of the FTA, that is, P(z) = 0 has a solution for any nonconstant polynomial P with complex coefficients, are suitable for a beginning course in Fourier transforms. Please see [3], [7], and [4] for basic results and an overview of the history of Fourier analysis. We shall use the following form of the Fourier transform and inversion formula, the proof of which is contained in the discussion found in [3, pp. 214–219]. Theorem 1 (FIF). Let f (x) : R → C be a continuous function. If ∞ −∞ | f (x)|dx < ∞, then the Fourier transform defined as F(t) = ∞ −∞ e −2πixt f (x)dx