Proving the FTA

Complex Analysis · Axiom Academy

EXAMPLE Proving the Fundamental Theorem of Algebra Use Liouville's Theorem to prove that every non-constant polynomial has a complex root Prove the Fundamental Theorem of Algebra : every non-constant polynomial has at least one root in . We argue by contradiction, using only one deep fact from complex analysis — Liouville's Theorem : a bounded entire function must be constant. You just proved the Fundamental Theorem of Algebra — an algebraic fact about polynomial roots — using analysis . Proof by contradiction: we assumed p(z) has no root, built a bounded entire function, and Liouville's Theorem forced a contradiction. Liouville's Theorem is the engine: a bounded entire function must be constant. It has no analogue in real analysis — is real-analytic and bounded yet not constant. Growth controls the bound: for the leading term dominates, so and at infinity — that decay is what makes f bounded. Algebra through analysis: this root-existence result is hard to prove with algebra alone, but complex analysis makes it short — a hallmark of the field's reach. The same Liouville machinery proves much more — it is one of the most powerful corollaries of Cauchy's theory.

This is the written version of the interactive lesson above. See the full Complex Analysis course.