The Fundamental Theorem of Algebra.

This theorem is also called the theorem of d'Alembert.

Theorem 6.3.1   Let be a non constant polynomial over . Then has a root.

Corollary 6.3.2   Let be a non constant polynomial of degree over . Then has exactly roots, counted with multiplicity.

First examples are displyed in subsection 7.

Corollary 6.3.3   Every non constant polynomial with real coefficients is the product of factors of degree 1 and 2.

Proof. Let be a polynomial with real coefficients. By thm 3.1, this polynomial has at least one root . If this root is real, then factors by .

Suppose that is not real. By thm 7.3, is also a root of . Thus, factors by    Re .

An example can be found in 7.6.

Noah Dana-Picard 2007-12-24