            Sessions 12.1   Fundamental Theorem of Algebra

Theorem 1. (Fundamental Theorem of Algebra). A non-constant polynomial has at least one root.

Proof. First note that , for otherwise would reduce to a constant. Now as and hence there is a positive number such that is bounded outside Moreover, is also bounded on being continuous on this compact subset of C and so is bounded on C. If has no roots, then is also entire and so, by Liouvilles Theorem, is constant and therefore is constant. This contradiction completes the proof of the theorem.

12.2 Other Applications

We can now easily prove a fascinating theorem about complex numbers.

Theorem 2.  Every non-constant complex polynomial of degree has exactly zeros.

Proof. For every positive integer and all complex numbers we know that Thus for where we get Multiplying by we get from which it follows that is divisible by .

Now, by the Fundamental Theorem of Algebra, there is a value such that So where is a polynomial of degree . By induction if is a polynomial of degree , then where and are constants. Consequently, every non-constant complex polynomial of degree has exactly zeros.

Theorem 3. Suppose that is a polynomial with real coefficients. If is a non-real zero of , then the conjugate of is also a zero of .

Proof. Suppose that is a polynomial with real coefficients. Since is a zero of By conjugation, Then and consequently, is a zero of . 