Math 444/544

Fall 2026

• Syllabus.
• Some resources if you want to learn TeX. Many students like Overleaf.
• Homework 1 (due 10/2):

  §1.1 #1, 2, 5, 6, 7, 9, 10. Optional: 11.

  Solutions: pdf, tex.

• Quiz 1 (10/7):

  1. Let a, b, q, r ∈ ℤ and b > 0. Define what we mean by ”when a is divided by b, the quotient is q and the remainder is r.&squo;

  §1.1 #6: Let a, b, c ∈ ℤ. Suppose that when a is divided by b, the quotient is q and the remainder is r, and that when q is divided by c, the quotient is k and the remainder is s. Prove that when a is divided by bc, the quotient is k.

• Homework 2 (due 10/9):

  §1.2 #1abc, 4b, 9, 12, 13, 17, 22, 25. Optional: 33.

  §1.3 #4, 18, 21.

  Solutions: pdf, tex.

• Quiz 2 (10/14):

  1. State Bézout's lemma (Theorem 1.2).

  §1.2 #25(a): If a, b, u, v ∈ ℤ are such that au + bv = 1, prove that (a,b) = 1.

  (b): Show by example that if au + bv = d > 1, then (a,b) may not be d.

• Homework 3 (due 10/16):

  §1.3 #25, 35a.

  §2.1 #1, 4, 6, 15, 20.

  §14.1 #8, 9. Optional: 19, 20.

  Solutions: pdf, tex.

• Quiz 3 (10/21):

  1. State Euclid's lemma (Theorem 1.5).

  §2.1 #20(a): if a² ≡ b² (mod n) then a ≡ ± b (mod n).

  (b): Same question when n is prime.

• Homework 4 (due 10/23):

  §3.1 #13, 27, 29, 31, 32. Optional: 44.

  §3.2 #1, 3.

  Solutions: pdf, tex.

• Quiz 4 (10/28):

  Based on §3.1 #32 and 31.

  1. Let R be a ring. Define the center Z(R).

  2. Prove that Z(R) is closed under addition.

  3. Let M2(ℝ) be ring of 2×2 matrices. Write down an element that is not in the center, with a proof.

• Homework 5 (due 10/30):

  Optional: Let ℕ = { 0, 1, 2, 3, … }, and define operations ⊕ and ⊙ by x ⊕ y = max(x,y) and x ⊙ y = x + y. Which of the ten axioms listed at the beginning of §3.1 (page 44) hold, and which ones fail? Explain any that are not obvious.

  §3.2 #15, 34, 35, 37. Optional: 39.

  §3.3 #8, 9, 25, 29.

  One more (not optional): Prove that the map f: ℤ[i] → ℤ given by f(a + bi) = a2 + b2 is not a ring homomorphism.

  Solutions: pdf, tex.

• Quiz 5 (11/24):

  1. Let R and S be rings, and φ:R → S be a map. Define what it means for φ to be a homomorphism, and for φ to be an isomorphism.

  §3.3 #25: Let L be the subring of M2(ℤ) given by lower triangular matrices. Show that the function f: L → ℤ given by f( a 0 b c ) = a is a surjective homomorphism but not an isomorphism.

• Homework 6 (due 11/6):

  §3.3 #27, 35. Optional: 42.

  §4.1 #1ad, 3, 5bd, 6, 11, 20.

  Optional: For any ring R, prove that the map φ: R[x] → R[x] defined by φ(a0 + a1x + a2x2 + … + anxn) = a0 − a1x + a2x2 − … + (-1)nanxn is an isomorphism.

  Solutions: pdf, tex.

• Homework 7 (due 11/13):

  §4.2 #5a, 6 (but only for 5a, not 5b–g).

  §4.4 #9, 10, 12.

  §5.2 #14a.

  §5.3 #2b. Optional: 9.