Note The Fundamental Theorem of Algebra: An Elementary and Direct Proof OSWALDO RIO BRANCO DE OLIVEIRA H H ere is a simple, differentiation-free, integration-free, trigonometry-free, direct, and elementary proof of the Fundamental Theorem of Algebra. One can ask, as soon as the complex numbers have been defined, whether every polynomial has a zero in the complex numbers. In this note I consider how early in the development of complex analysis this question can be answered. As pointed out by Remmert [11], Burckel [2], and others, two of the best analytical proofs of the FTA are the easy and short but not elementary one given by Argand [1] (see also [3, 4, 6, 7, 10, 11, 12, 14]), and the elementary but not so easy or short one given by Littlewood [9] (see also [5, 8, 11, 13]). All these works, except [2], use or prove d’Alembert’s Lemma [14] or Argand’s Inequality [11]: ‘‘If P is a nonconstant complex polynomial and Pðz 0 Þ 6¼ 0; z 0 2 C, then any neighborhood of z 0 con- tains a point w such that |P(w)| \ |P(z 0 )|’’. The proof of the FTA I will now present does not apply d’Alembert’s Lemma or Argand’s Inequality. Instead I assume without proof only the continuity of complex polynomials and the following consequences of the com- pleteness of R: • Any continuous function f : D ! R; D a bounded and closed disc, has a minimum on D. • Every positive real number has a positive square root. Square Roots. It is well known that z 2 ¼ a þ ib; a; b 2 R, is solvable in C. We have z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 þ ffiffiffiffiffiffiffiffiffi a 2 þb 2 p 2 q þ isgnðbÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  a 2 þ ffiffiffiffiffiffiffiffiffi a 2 þb 2 p 2 q , with sgnðbÞ¼ b jbj , if b = 0, and sgn(0) = 1. Applying this formula repeatedly we find all the 2 j -roots, j 2 N, of z =±1 and z =±i. Fundamental Theorem of Algebra. Let P be a complex polynomial, with degreeðPÞ¼ n  1. Then there exists z 0 2 C satisfying P(z 0 ) = 0. PROOF. Writing P ðzÞ¼ a 0 þ a 1 z::: þ a n z n , with a j 2 C; 0  j  n; a n 6¼ 0, we have jP ðzÞj  ja n jjzj n ja 0 j ::: ja n1 jjzj n1 ; from which follows lim jzj!1 jP ðzÞj ¼ 1. By continuity, it is easy to see that |P| has an absolute minimum at some z 0 2 C. Suppose without loss of generality that z 0 = 0. Hence, putting S 1 ¼fx 2 C : jxj¼ 1g; jPðr xÞj 2 jP ð0Þj 2  0; 8r  0; 8x 2 S 1 ; ð1Þ and P(z) = P(0) + z k Q(z), for some k 2f1; :::; ng, where Q is a polynomial and Q(0) = 0. Substituting this expression, Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 2, 2011 1 DOI 10.1007/s00283-011-9199-2