436 CHAPTER 15. DEGREE THEORY

17. Using Problem 16, prove the fundamental theorem of algebra as follows. Let p(z)be a nonconstant polynomial of degree n, p(z) = anzn + an−1zn−1 + · · · Show thatfor large enough r, |p(z)|> |p(z)−anzn| for all z ∈ ∂B(0,r). Now from Problem 15you can conclude d (p,Br,0) = d ( f ,Br,0) = n where f (z) = anzn.

18. Suppose f :Rp→Rp satisfies |f (x)−f (y)| ≥α |x−y| , α > 0. Show that f mustmap Rp onto Rp. Hint: First show f is one to one. Then use invariance of domain.Next show, using the inequality, that the points not in f (Rp) must form an open setbecause if y is such a point, then there can be no sequence {f (xn)} converging toit. Finally recall that Rp is connected.

19. Suppose D is a nonempty bounded open set in Rp and suppose f : D→ ∂D is con-tinuous with f (x) = x for x ∈ ∂D. Show this cannot happen. Hint: Let y ∈ Dand note that id and f agree on ∂D. Therefore, from properties of the degree,d (f ,D,y) = d (id,D,y). Explain why this cannot occur.

20. Assume D is a closed ball in Rp and suppose f : D→ D is continuous. Use theabove problem to conclude f has a fixed point. Hint: If no fixed point, let g (x) bethe point on ∂D which results from extending the ray starting at f (x) to x. Thiswould be a continuous map from D to ∂D which does not move any point on ∂D.Draw a picture. This may be the easiest proof of the Brouwer fixed point theorembut note how dependent it is on the properties of the degree.

21. Use Corollary 15.6.9 to prove the invariance of domain theorem that if U is openand f : U ⊆ Rp → Rp is continuous and one to one, then f (U) is open. This wasdiscussed in the chapter but go through the details.