Reading:

Read handout and Chapters 7 & 10 from course text.

Problems:

From course text:

Chapter 10: 10.4, 10.5

Problem A: Let D=(P,B) be an affine plane of order n. Show that there are n lines in any parallel class. Conclude that there are that there are n+1 classes of parallel lines.

Problem B: Let D be the affine plane of order 5 coming from F_5 x F_5. Draw the points of the D as a 5 x 5 grid with rows and columns labeled by F_5. Draw the line given by y=2x+3.

Problem C: Let D be the affine plane of order 3 coming from F_3 x F_3. Drawing the points as a grid as in the previous problem, draw all lines of D.

Problem D: Using the last problem, draw a picture of the projective plane of order 3.

Problem E: Write down a homogenous equation for y=x^2+3x+1. By solving the homogenous equation, show there is only one point at infinity (give its coordinates). This shows both branches of the parabola y=x^2+3x+1 meet at infinity, justifying our geometric intuition that the parabola becomes a loop projectively.