Theorems

Fundamental Theorem of Algebra

Also Known As d'Alembert-Gauss Theorem

Algebra

Citation Formats

General Reference

Mathematics Atlas. "Fundamental Theorem of Algebra." Accessed August 29, 2026. https://dev-mathematics.interactiveatlas.org/theorems/fundamental-theorem-of-algebra.

APA Style

Mathematics Atlas. (n.d.). Fundamental Theorem of Algebra. Retrieved August 29, 2026, from https://dev-mathematics.interactiveatlas.org/theorems/fundamental-theorem-of-algebra

BibTeX

@misc{mathematicsatlas-fundamental-theorem-of-algebra,
  author = {Mathematics Atlas},
  title = {Fundamental Theorem of Algebra},
  year = {2026},
  url = {https://dev-mathematics.interactiveatlas.org/theorems/fundamental-theorem-of-algebra},
  note = {Accessed August 29, 2026}
}

Guarantees that the complex numbers are, in a precise sense, complete enough for algebra: no equation solvable in principle needs a still larger number system than the complex numbers to find its roots. Several mathematicians, including d'Alembert and Euler, attempted proofs earlier in the eighteenth century, but each had gaps; Carl Friedrich Gauss's 1799 doctoral dissertation gave the first substantially rigorous proof, and he returned to give three further, increasingly rigorous proofs over his career.

Facts

Statement

Every non-constant single-variable polynomial with complex coefficients has at least one root in the complex numbers. 1

Proof Year

1799 1

Gauss's 1799 doctoral dissertation gave the first substantially rigorous proof; earlier eighteenth century attempts by d'Alembert (1746) and Euler had gaps a fully rigorous treatment had to close.

Cross-Tradition Connections