Let be a field contained in , and fix a number in . Recall that is the projective plane, while is the sphere of radius :
This assignment is meant to show is not the same as . This assignment is due Friday, September 6, 2013 at the start of class.
Problem 1. For each point satisfying , consider the line
Define a map by considering where the line intersects the sphere :
Show that is a well-defined map.
Problem 2. Show that is surjective. Hint: Show if and only if .
Problem 3. Verify the following set equalities and isomorphisms:
Conclude that .
Problem 4. Show that is injective on , i.e., if then .
Problem 5. Show that is not injective on . Conclude that is not the same as .