Math 332 Assignments

General remarks

Assignment 1, due September 7

Read Chapter 1 of Adler and Coury.

Do the following problems:
pp. 32-33: 1, 6, 18, 19, 23, 31, 36.

Assignment 2, due September 14

Do the following problems:
pp. 32-35: 7, 11, 30, 38, 40, 42, 50.

Assignment 3, due September 21

Read pp. 357-362 of Adler and Coury.

Do the following problems:
p. 387: 1, 6, 7, 15.
p. 35: 58.

Let s be a square root of -2. Let Z[s] = [a+bs : a, b integers}.
Z[s] has a norm function N(a+bs) = (a+bs)(a-bs),
and this norm gives Z[s] division with remainder,
and a Euclidean algortihm. Z[s] has unique factorization.

A1. Find all the units of Z[s].

A2. Find some ordinary (integer) primes which are Z[s]-primes
and some ordinary primes which are not Z[s]-primes. Find at
least six of each and explain how you decide.

Conjecture how to decide whether an ordinary integer
prime is a Z[s]-prime.

Assignment 4, due September 28

Read Chapter 2 of Adler and Coury.
Chapters 5,6, 8 of Childs may be helpful additional reference.
Adler and Coury p. 64 et seq.: 3, 10, 13, 15, 17, 38, 41.

Assignment 5, due October 5

Read Chapter 3 of Adler and Coury.

Do the following problems:
p. 64 et seq.: 29, 33, 36.
p. 94 et seq.: 13, 22, 31, 32.

Assignment 6, due October 12

Read Chapter 4 of Adler and Coury.

Do the following problems:
Adler and Coury p. 122-123: 1, 6, 13, 16, 17, 29.

B1. Let a, b, p be natural numbers, such that p is prime and a + b = p - 1.
Show a!b! + (-1)^a is congruent to 0 modulo p. (from Davenport, Higher Arithmetic)

Assignment 7, due October 26

Read Chapter 6 of Adler and Coury.

Do the following problems:
Adler and Coury p. 189 et seq.: 3, 12, 15, 28, 37, 40, 41.

Assignment 8, due November 2

Read Chapter 5 of Adler and Coury.

Do the following problems:
Adler and Coury p. 153 et seq.: 6, 8, 18, 30, 42, 46, 49.

Back to course home page.

Assignment 9, due November 9

Read Chapter 9 of Adler and Coury.

Do the following problems:
Adler and Coury p. 308 et seq.: 1, 5, 11, 12b, 17, 24, 30.

Assignment 10, due November 26

Read Chapter 10 of Adler and Coury.

Do the following problems:
Adler and Coury p. 263: 1, 5.
p. 352: 3, 9,12, 13, 19.

Assignment 11, due December 3

Read Chapter 8 and pp. 200-201 of Adler and Coury.

Do the following problems: (Note. Sketches may help your intuition.)

K1. Let r be a positive rational number.
Let C be the unit circle with the graph x² + y² = 1.
Let L be the line through the point (-1,0) with slope r.

Show: L intersects C at a second point with rational
coordinates (x/z, y/z), and (x,y,z) is a Pythagorean triple.

K2. Let C be the unit circle. Let L be any line with
rational slope which intersects C at two distinct points P and Q.

Show: if P has rational coordinates, then Q has rational coordinates.

K3. Let E be the curve with the graph x³ + y³ = 9.
Let L be the line x + 4y = 9.
Line L is tangent to E at the point (1,2) and intersects E at another point P.

Find P. (i.e. what are its coordinates?)

K4. Let E be the same curve as in problem K3.
Let M be the line through the point (2,1) and P. (Corrected)
Line M intersects E at a third point Q.

Show Q has rational coordinates.

K5. Let S be the curve with the graph x⁴ + y⁴ = 1.

Find all of the points on S with rational coordinates.

K6. Let T be a set of N positive integers whose prime divisors are less than 20.

What is the smallest value of N which guarantees some nonempty subset of T has product which is a perfect square?