Homework 8 for MATH 185
Due: Wednesday March 21, 3:10 pm in class
Problem 1
Give two new proofs of the fundamental theorem of algebra:
(a) Using the open mapping theorem. (Show that the image P(C)of a complex polynomial Pis open
and closed.)
(b) Using the mininmal modulus principle.
Problem 2
Let Tbe a Möbius transformation, i.e. Tis of the form
T(z) = az +b
cz +d,
where a,b,c,d∈Csuch that ad −bc 6=0.
(a) Show that Tcan be written as a composition T=T4◦T3◦T2◦T1, where T1and T4are translations,
T2is an inversion, and T3is a rotation-dilation.
(b) Use part (a) to show that if L⊂Cis a straight line and S⊂Cis a circle, then T(L)is either a
straight line or a circle, and T(S)is either a straight line or a circle.
