Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has a root among the complex numbers.

Specifications & Metrics

proved

Gauss 1799 (first accepted proof); fully rigorous proofs later

Core Principles & Analysis

• Gauss's dissertation proof contained a topological gap, but convinced everyone anyway.
• Every proof uses analysis or topology — the theorem is algebraic in statement only.
• It is why complex numbers cannot be merely optional: without them, polynomial algebra breaks.