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.