Homework for Math 540 : Spring 2017


The sections referred to in the problems below come from the online text by Stein or Clark.

HW #1: Clark, 2.5, 2.6, 2.7, 2.9. 2.10. 2.11 (due 1/27/17).

HW #2: Clark, 7.3, 9.1, 9.2, 9.3. For 9.1, use the Euclidean algorithm and reverse substitution (due 1/27/17).

HW #3: Clark, 9.1, this time using Blankinship's method (due 1/27/17).

HW #4: Stein, 1.3, 1.8, 1.12, 1.13 (due 2/3/17).

HW #5: Stein, 1.7, 1.14 (due 2/3/17).

HW #6: Clark, 13.2, 13.3, 13.4 (due 2/10/17).

HW #7: #1. Let S denote the set of real numbers. For a, b in S, define a ~ b to mean a-b is in Z. Prove that ~ is an equivalence relation.
#2. Let T denote the ordered pairs of integers (a, b), with b non-zero. For (a, b) and (c, d) in T, define (a, b) ~ (c, d) to mean ad=bc. Prove:

  1. ~ is an equivalence relation.
  2. If (a, b) ~ (a', b') and (c, d) ~ (c', d'), then (ad+bc, bd) ~ (a'd'+b'c', b'd').
  3. If (a, b) ~ (a', b') and (c,d) ~ (c', d'), then (ac, bd) ~ (a'c', b'd').
Note that the construction in #2 gives rise to the formal definition of the rational numbers: They are, by definition, the equivalence classes obtained from the relation in #2. (due 2/17/17).

HW #8: Clark, 15.4, 15.5, 15.6, 15.7, 15.8 (due 2/17/17).

HW #9: #1. Write out addition and multiplication tables for Z4 and Z12. #2. Consider the equations 3x ≡ 2 and 2x ≡ 10. Use #1 to find all solutions to these equivalences (separately) in both Z4 and Z12, (due 2/17/17).

HW #10: Due 2/24/17.

  1. Prove that for positive integers b,c and a prime number p, (b+c)p ≡ bp + cp (mod p). Hint: Use the binomial theorem.
  2. Find a solution to the system of congruences: x ≡ 4 (mod 6), x ≡ 6 (mod 7), and x ≡ 10 (mod 13).
  3. Show that your answer in #2 is unique, modulo 546.

HW #11: Let φ(n) denote the Euler phi function. Use the formulas from class to derive the following (due 2/24/17).

  1. φ(ab) = φ(a)φ(b) · (d ÷ φ(d)), where d = GCD(a,b).
  2. If a|b, then φ(a) | φ(b).

HW #12: Stein, 2.8, 2.13, 2.23 (due 3/3/17).

HW #13: Stein, 2.5, 2.10, 2.11, 2.14. Note for 2.11, you are being asked to find the least positive r for which 2r is congruent to 1 modulo 17 (due 3/3/17).

HW #14: Stein, 2.24, 2.26, 2.27(a) (due 3/10/17).

HW #15: Stein 2.33 and the following two problems, for which p denotes a prime number (due 3/10/17).

  1. Let f(x) = adxd + · · · + a0 and g(x) = bdxd + · · · + b0 belong to Z[x]. Suppose ai ≡ bi (mod p), for all i. Prove that an integer u is a root of f(x) modulo p if and only if u is a root of g(x) modulo p.
  2. Let f(x) = ax2+bx+c belong to Z[x] and assume a is not divisible by 2. Find all roots of f(x) modulo 2. Hint: Use the previous problem.

HW #16: Stein, 4.1, 4.8 and the following: Use Euler's criterion to find the quadratic residues modulo 19 and then find both square roots of these residues modulo 19 using the fact that 19 ≡ 3 (mod 4). (due 3/17/17)

HW #17: In the problems below, (a/p) denotes the Legendre symbol of a over p (due 3/17/17).

  1. Use the law of quadratic reciprocity and properties of the Legendre symbol to calculate (-42/61), (-1/61), (2/61), (7/61).
  2. Use the law of quadratic reciprocity to determine if 3x2+10x-11 has a root modulo 227. You need not find a root, if a root exists modulo 227.
  3. Find the prime numbers p less than 50 for which (5/p) = 1.

HW #18: Use Gauss's Lemma to calcluate: (2/17), (3/11) and (9/13). All details must be provided. (due 3/31/17)

HW #19: Stein 4.3, and the following: Use Gauss's Lemma to prove (2/p) = -1, if p ≡ 5 (mod 8). (due 3/31/17)

HW #20: For the two pairs of Guassian integers, (2-12i, 6+8i) and (24-i, 36+24i), find a GCD for each pair, and express that GCD as a linear combination of the pair, as guaranteed by Bezout's Principle. (due 4/21/17)

HW #21: Factor each of the following Gaussian integers as a product of Gaussian primes: 7350, 3+6i, 1+5i. (due 4/21/17)

HW #22: For Gaussian integers x= 6+8i, y = 3+11i, and z= 1+2i find Gaussian integers x', y' such that x ≡ x' (mod z) and y ≡ y' (mod z), with N(x') < N(z) and N(y') < N(z). Then verify x+y ≡ x'+y' (mod z) and xy ≡ x'y' (mod z). (due 4/28/17)

HW #23: For the Gaussian priime 2+i, find a distinct set of residue classes modulo 2+i, and for the four non-zero classes, find the order of each class modulo 2+i. (due 4/28/17)

HW #24: Solve the following congruences over the Gassian integers: (a) (3+5i)x ≡ 7-i (mod 3-5i) and (b) (1+i)x ≡ 10-3i (mod 3+2i). (due 5/3/17)

HW #25: For the Gaussian integers z = 2+3i and y = 3-i, find a full set of distinct residue classes, so that the norm of each representative of the class is less than the norm of the Gaussian integer defining the class. (due 5/3/17)