# Math 391

## Fall 2020

• Syllabus.
• Resources if you want to learn TeX.
• Handout on writing by Jack Lee.
• Lecture 1: Welcome; binomial theorem.
notes, worksheet.
• Lecture 2: Greatest common divisors.
notes, worksheet.
• Reading 1 (due 10/4): §1.2 and §1.3 through page 24 (stop after the proof of Proposition 3.4).
• Homework 1 (due 10/5):
§1.1 #4 (choose four parts), 5, 7.
Solutions: pdf, tex.
• Lecture 3: The Euclidean algorithm.
notes, worksheet.
• Lecture 4: Modular arithmetic.
notes, worksheet.
• Lecture 5: Divisibility criteria.
notes, worksheet.
• Homework 2 (due 10/12):
§1.2 #2, 5, 11, 13.
§1.3 #15.
Solutions: pdf, tex.
• Lecture 6: Fermat's little theorem.
notes, worksheet.
• Lecture 7: Solving congruences, I.
notes, worksheet.
• Lecture 8: Solving congruences, II.
notes, worksheet.
• Reading 3 (due 10/18): §1.4
• Homework 3 (due 10/19):
§1.3 #14, 21, 23.
Solutions: pdf, tex.
• Lecture 9: RSA cryptography.
notes, worksheet.
• Lecture 10: Rings.
notes, worksheet.
• Lecture 11: Units and zero-divisors.
notes, worksheet.
• Reading 4 (due 10/25): §2.1 and §2.2.
• Homework 4 (due 10/28): §1.4 #11, 4, 19bd, and:
• Let R be a commutative ring, fix an element r ∈ R, and consider the map R → R given by x ↦ rx. Show that this map is injective if and only if r is neither zero nor a zero-divisor, and that it is surjective if and only if r is a unit.
• RSA problem, building on Worksheet 9:
Write a message you want to send me, encode it in blocks of nine letters using Shifrin's code, encrypt it using the public key N = 340196134436655019 and e = 3, and email it to me. Also send me your public key. I'll send back the plain text of your message, and an encrypted message for you.
Bonus points if my computer can't factor the number N in your public key in less than a minute. But no bonus points if I can google your number.
Solutions: pdf, tex.
• Lecture 12: The ring ℤm.
notes, no worksheet.
• Lecture 13: Rational numbers, ordered fields.
notes, worksheet.
• Lecture 14: Real numbers.
notes, worksheet.
• Reading 5 (due 11/1): §2.3 and §2.4.
• Homework 5 (due 11/6):
Write up Worksheet 10 or Worksheet 12 (from Lecture 13).
Solutions to Worksheet 10: pdf, tex.
Solutions to Worksheet 12: pdf, tex.
• Lecture 15: Complex numbers.
notes, worksheet.
• Lecture 16: Roots of unity.
notes, worksheet.
• Lecture 17: Solving cubic polynomials.
notes, worksheet.
• Reading 6 (due 11/8): §3.1 and §3.2, but skip Descartes’ rule of signs if you want.
• Lecture 18: Polynomial rings.
notes, no worksheet.
• Lecture 19: GCDs and the Euclidean Algorithm.
notes, worksheet.
• Homework 6 (due 11/13):
§2.3 #6, 10, 15.
§2.4 #1 (choose two parts), 6 (but ignore the thing about Δ).
Solutions: pdf, tex.
• Lecture 20: Irreducible polynomials.
notes, worksheet.
• Lecture 21: Unique factorization; roots.
notes, worksheet.
• Lecture 22: Rational root theorem.
notes, worksheet.
• Homework 7 (due 11/20):
§3.1 #1de, 8, 9, 10.
Solutions: pdf, tex.
• Lecture 23: Irreducibles in ℤp[x].
notes, worksheet.
• Reading 7 (due 11/22): §3.3.
• Lecture 24: Gauss’s lemma, Eisenstein’s criterion.
notes, no worksheet.
• Lecture 25: Subrings, isomorphisms.
notes, worksheet.
• Reading 8 (due 11/29): §4.1.
• Homework 8 (due 11/30):
§3.3 #2: do half, and use the table on page 112 if you want.
By the rational root theorem (Proposition 3.1), the polynomial x2-2 has no roots in ℚ. Spell out the proof of the theorem in this example. Is it very different from your proof that √2 is irrational in §1.2 #11, or is it more or less the same? Explain.
Challenge: §3.3 #10.
Solutions: pdf, tex.
• Lecture 26: Homomorphisms.
notes, worksheet.
• Lecture 27: Ideals.
notes, no worksheet.
• Lecture 28: A non-principal ideal.
notes, worksheet.