The Fundamental Theorem of Algebra via Linear Algebra

Theorem 2 is also consequence of Theorem 1, so the two theorems are equivalent. In fact, the implication Theorem 1 ⇒ Theorem 2 is usually how one first meets the fundamental theorem of algebra in a linear algebra course: it assures us that any complex square matrix has an eigenvector because the characteristic polynomial of the matrix has a complex root. But here, we will prove Theorem 2 without assuming Theorem 1, so we can deduce Theorem 1 as a consequence of Theorem 2. Our argument is a modification of a proof by H. Derksen [1]. It uses an interesting induction. Our starting point is the following lemma.