The Fundamental Theorem of Algebra and Linear Algebra

Abstract

The first widely accepted proof of the Fundamental Theorem of Algebra was published by Gaus in 1799 in his Ph.D. thesis, although to current standards this proof has gaps. Argand gave a proof (with only small gaps) in 1814 which was based on a flawed proof of d’Alembert of 1746. Many more proofs followed, including three more proofs by Gaus. For a more about the history of the Fundamental Theorem of Algebra, see [5, 6]. Proofs roughly can be divided up in three categories (see [3] for a collection of proofs). First there are the topological proofs (see [1, 8]). These proofs are based on topological considerations such as the winding number of a curve in R around 0. Gaus’ original proof might fit in this category as well. Then there are analytical proofs (see [9]) which are related to Liouville’s result that an entire non-constant function on C is unbounded. Finally there are the algebraic proofs (see [4, 10]). These proofs only use the facts that every odd polynomial with real coefficients has a real root, and that every complex number has a square root. The deeper reasons why these proves work can be understood in terms of Galois Theory. For a linear algebra course, the Fundamental Theorem of Algebra is needed, so it is therefore desireable to have a proof of it in terms of linear algebra. In this paper we will prove that every square matrix with complex coefficients has an eigenvector. This is equivalent to the Fundamental Theorem of Algebra. In fact we will prove the slightly stronger result that any number of commuting square matrices with complex entries will have a common eigenvector. The proof is entirely within the framework of linear algebra, and unlike most other algebraic proves of the Fundamental Theorem of Algebra, it does not require Galois Theory or splitting fields. Another (but longer) proof using linear algebra can be found in [7].