Let $p:\mathbb C\longrightarrow \mathbb C$ be a complex polynomial with no zeros and degree $n$. Is it true that the maps $f, g:S^1\longrightarrow S^1$ given by $$f(z)=z^n\quad \textrm{and}\quad g(x)=\frac{p(z)}{|p(z)|}$$ are homotopic?
Obs: I'm supposing $p$ has no zeros for finding a contradiction for a proof of the fundamental theorem of algebra.