UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 1

Problem:

Solution:

Theorem 7 is the fundamental theorem of algebra, staying that any polynomial with coefficients in $\mathbf{C}$ has a root in $\mathbf{C}$.

Now suppose $f$ has degree $n$ with coefficients in $\mathbf{C}$, but the fundamental theorem of algebra we have $f$ have a root in $\mathbf{C}$, so using the division theorem we know $f = (x - a)g(x)$ with $g(x)$ a polynomial of degree $n - 1$.

By induction we get the result we want.