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.

Homework Assignment 1 Download

**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 .

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