Homework is to be collected in class on the dates indicated.

Homework #1: (due Friday September 12)

Rudin Chapter 1 problems 1, 2, 3, 4, 6, 7, 9, 20

A. Show that the set {0, 1, 2, ..., p-1}
with addition and multiplication defined mod p is a field. Here p is prime.
Does it have to be?

B. Show that any finite set can be made into an ordered set.

C. Does the field of rational numbers have any subfields?

NOTE: PROBLEMS 1, 6, 7, 9, 20 REQUIRE THE REAL NUMBERS. WE WILL
COVER THE MATERIAL ON WEDNESDAY. THEY CAN BE
HANDED IN on Monday, September 15.

Homework #2: (due Friday September 19)

Rudin Chapter 2 problems 2, 3, 4

A. In class, we showed, using the properties of the real numbers, together
with certain ``geometric axioms'' regarding areas, that given two circles, the ratio of their
areas is as the square of the ratio of their diameters. In this problem, you are asked
to formulate analogous axioms for ``length'', and prove a well-known
relation between the area and circumfurence of a circle.

HINT: One of the axioms should certainly be that arcs of circles have
a  real number ``length''.

Homework #3: (due Monday September 29)

Rudin Chapter 2 problems  6, 8, 10, 12, 13, 14, 15, 16, 17, 18, 22, 24, 25, 26, 27, 28, 29

A. Show, using a Cantor's diagonalization process, that the real numbers,
as we have defined them, are uncountable.

Homework #4: (due Friday October 17)

Rudin Chapter 3 problems  1, 20, 21, 23, 24, 25

Homework #5: (due Monday October 27)

Rudin Chapter 3 problems  2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 19, 22

A. In class we defined the extended real numbers R* as a set
containing the real numbers. Show that it can be made into an ordered
set, such that the induced ordering on its subset R coincides with the
usual ordering on R.

B. Can R* be made into an ordered field containing R (with its usual
field operations) as a subfield?

Homework #6: (due Monday November 3)

Rudin Chapter 4 problems  2, 3, 4, 6, 9, 10, 14, 18

Homework #7: (due Friday November 7)

Rudin Chapter 4 problems  19, 22, 26

Homework #8: (due Wednesday November 26)

Rudin Chapter 5 problems  1, 2, 4, 5, 6, 11, 12, 13, 15, 16, 22, 23, 24, 25, 26, 27, 28, 29

Homework #9: (due last day homework can be due)

Rudin Chapter 6 problems  1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 15, 16, 17