Complex Polynomials and the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra (FTA)

Theorem

For all with , there exists a such that .

That is, when , every polynomial of positive degree has at least 1 root.

Complex Polynomials of Degree n Have n Roots (CPN)

Proposition

For all integers , and all complex polynomials of degree , there exists complex numbers and such that:

and the roots are

Proof

We will prove this by induction on , where is the open sentence:

For all complex polynomials of degree , there exist complex numbers and such that:

and the roots of are

Base Case:
For , we have for complex numbers and with .
We can write where and .
Now, for , and thus since , is a root of .
That is, is the one and only root of .
Thus is true.

Inductive Step: let be an arbitrary natural number.
Assume the inductive hypothesis, .
We wish to prove .

Consider a complex polynomial of degree .
By FTA, has a complex root, which we will call .
By FT, we know that for some complex polynomial .
This quotient must have degree by DAP.
Thus, by the inductive hypothesis, there exist complex numbers and such that:

and the roots of are .
Substituting, we obtain:

Since , a complex number is a root of if and only if .
That is, the roots of are precisely and , and the roots of .

The result if true for and hence holds true for all by POMI.