The Fundamental Theorem of Algebra

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.

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 .