Algebraic Topology · Problem 2.2.8

Homology — Allen Hatcher

Mathematics — Topology · degree-theory

Problem

A polynomial

f(z)

with complex coefficients, viewed as a map

\mathbb{C} \to \mathbb{C}

, can always be extended to a continuous map of one-point compactifications

\hat{f}: S^2 \to S^2

. Show that the degree of

\hat{f}

equals the degree of

f

as a polynomial. Show also that the local degree of

\hat{f}

at a root of

f

is the multiplicity of the root.