Counting roots of polynomial inside $S^1$ I would like to ask for a hint to this problem:
Let $p$ a polynomial function on $C$ with no root on $S^1$. Show that the number of roots of $p$ with $|z|<1$ is the degree of the map $q: S^1 \to S^1$ given by $ q(z)=p(z)/|p(z)|$.
This problem appears in Peter May's book, concise course in Algebraic Topology. Assuming the fundamental theorem of algebra, my idea so far is to collapse every zero inside $S^1$  through a homotopy to zero, explicitly, if $a_1, ..., a_k$ are the roots inside $S^1$  take $h(z,t)=(z-t*a_1)...(z-t*a_k)*p_2(z)$, where $p_2(z)$ is the part of the polynomial that has the zeros outside $S^1$, and consider $h(z,t)/|h(z,t)|$. Then, I end up with something like $z^k*p_2(z)/|p_2(z)|$. Next, I think of vanishing, somehow, the part involving $p_2$, but don't know how.
Any advice would be appreciated.
 A: Here is an idea (caution, I did not try this but it is what I would try if I was attempting the question):
The degree of a continuous map $S^n \to S^n$ is called the winding number if $n=1$ and it may be computed using the following formula:
$$\text{ winding number} = {1 \over 2 \pi i}\oint_C {dz \over z}$$
where $C$ is a curve around $0$. In your question $C = q(z)$ and therefore the winding number of $q$ becomes
$$\text{ winding number of q} = {1 \over 2 \pi i}\oint_{q \circ \gamma} {dz \over z}$$
where $\gamma(t) = e^{2 \pi i t}$ and $t \in [0,1]$. 
So now we have the degree of $q$. It remains for us to count the roots of $p$. To this end, note that if $p$ has a root at $z_0$ then ${1 \over p}$ has a pole at $z_0$. To count the poles of a map inside a closed curve you may use the residue theorem:
Let $\beta$ be any curve inside $S^1$ and outside all roots of $p$. Then by the residue theorem
$$ \oint_\beta {1 \over z} dz = 2\pi i \sum_{k=1}^n \mathrm{Res}({1\over p}, a_k)$$
where $a_k$ are the roots of $p$. Now all that remains to be done is to calculate the residues and then show that the two are equal.
