# Math 3360: Applicable Algebra, Spring 2018 Prof. Allen Knutson The final exam, with answers

Location: Malott 207
Class time: 10:10-11:00 (5482), 11:15-12:05 (5483)
Text: [Childs] "A Concrete Introduction to Higher Algebra"
Office hours: Wednesday 12:05-1 for now at least
TA office hours: Rakvi Christine (5482) on Monday 2-4 in Malott 114, TaoRan Chen (5483) on Tuesday 2-4 in Malott 218. At other times you can try the Mathematics Support Center in Malott 256.
Prelims: 2/22, 4/19 evening 7:30 PM in Goldwin Smith G76
Final exam: Tuesday, May 15, 9:00 AM

Course description:

Introduction to the concepts and methods of abstract algebra and number theory that are of interest in applications. Covers the basic theory of groups, rings and fields and their applications to such areas as public-key cryptography, error-correcting codes, parallel computing, and experimental designs. Applications include the RSA cryptosystem and use of finite fields to construct error-correcting codes and Latin squares. Topics include elementary number theory, Euclidean algorithm, prime factorization, congruences, theorems of Fermat and Euler, elementary group theory, Chinese remainder theorem, factorization in the ring of polynomials, and classification of finite fields.
Homework will be due Fridays at the beginning of class and count 30% of the final grade. Prelims each 20%, final exam 30%.

Initial guess at the topics and timeline. All chapters refer to [Childs]. * indicates a break.

• Jan 24: One-to-one functions. Image of a function. Inclusion of a subset. Every function factors uniquely through the inclusion of its image.
• Jan 26: Onto functions, partitions of a set.
• Jan 29: Relations, functions as relations, equivalence relations. Integers as equivalence classes, rationals as equivalence classes.
• Jan 31: Proofs by case analysis, contradiction, induction.
• Feb 2: Some induction proofs: Fibonacci formula, (x choose k) is an integer-valued polynomial (via Pascal + induction), an integer-valued polynomial is an integer combination of (x choose k)s.
• Feb 5: Total orderings on sets, well-orderings. Bonus topic: juggling patterns, whose average we proved to be the number of balls. Division theorem.
• Feb 7: Continued fractions. Euclid's algorithm.
• Feb 9: Prime vs. irreducible, e.g. in ZZ[sqrt(-5)]. Fundamental theorem of arithmetic. Corollary: lcm(a,b) = ab/gcd(a,b).
• Feb 12: congruence is/as an equivalence relation. If a = r mod x and mod y, then a = r mod lcm(x,y).
• Feb 14: equivalence relations used in topology, to make Asteroids on a torus and Valentine-related tomfoolery. Fermat's Little Theorem. Casting out 9s, 99s, 999s. Solving ax=b mod m when gcd(a,m)=1.
• Feb 16: solving ax = b mod m for general a,b,m
• Feb 21-23
• Feb 26: Bruhat decomposition of matrices. Matrices are invertible iff they reduce (by downward row and rightward column operations) to permutation matrices, hence must be square. Rank of a matrix.
• *Feb 26-Mar 2
• Mar 5: Determinants and Sam Loyd's 14-15 problem. Projective planes and Spot It!. Linear algebra over F_2 and Lights Out.
• Mar 5-9
• Mar 7: Euler's generalization of Fermat's Little Theorem, RSA encryption, Heninger et al.'s absurdly successful attack on RSA-encrypted websites.
"Dracula encrypts himself for security" -- Matt Levine
• Mar 9: Image of a group homomorphism, subgroups, cosets.
• Mar 12: Lagrange's theorem. Orders of elements. G/H might not be a group (such that G -> G/H is a homomorphism), but if G is abelian then it is. Simple groups.
• Mar 12-16
• Mar 19-23
• Mar 26: CRT for polynomials. If f(x) = 3 mod x^2, and f(x) = 5 mod (x-2)^2, use Euclid->Bezout to compute f(x) mod (x(x-2))^2.
• *Apr 9-13
• Apr 16-20
• Apr 23-27
• Apr 27: Two field elements related by a field automorphism have the same minimal polynomial. Aut(F_{p^n}) is generated by the Frobenius. Classification of subfields of finite fields. Compuing inverses in finite fields, using inverses of matrices. Counting irreducible polynomials using inclusion-exclusion on the field elements.
• Apr 30-May 4
• May 7-9
• HW #1, due Friday Feb 2:
• 1. Give an example of functions f,g such that g o f is 1:1 & onto, but f isn't onto and g isn't 1:1.
• 2. Find the smallest example (the smallest three sets) and prove that yours is indeed the smallest example.
• 3. Count the number of partitions of an n-element set, for n=0,1,2,3,4. Then feed into The On-Line Encyclopedia of Integer Sequences to find out the names of these numbers.
• 4. Let f:A->B be a function, and q:A->A/f be the quotient map taking a |-> f-1(f(a)). Prove that there exists a unique function g:A/f->B such that f = g o q.
• HW #2, due Friday Feb 9:
• Exercise 2 from chapter 1 of [Childs].
• If a relation is transitive and symmetric, does that imply that it is reflexive? Either prove "yes that's implied", or give a counterexample. "It doesn't seem that you can derive it" isn't good enough -- only an explicit "here's a set and a relation that's T & S but not R" is good enough.
• Chapter 2 exercises 2,8,26,37.
• HW #3, due Friday Feb 16:
• Chapter 3 exercises 27, 28, 31, 53, 72.
• Chapter 4 exercise 8.
• Feb 22 is the first prelim. Some study problems:
• I.2: 3, 5, 8, 15, 23, 24, 30, 31 is cool but too tricksy for a test
• I.3: 7, 18, 25, 33, 36, 40, 46, 50, 51, 63, 67
• I.4: 2, 12, 17, 19, 23, 27, 28, 29, 39 (here [] means LCM), 43
• I.5: 6, 7, 13, 15, 21, 23, 24, 27, 33, 49, 53
• HW #4, due Friday Mar 2:
• Chapter 4 exercise 43
• Chapter 5 exercises 12, 46, 54
• Chapter 6 exercise 58
• Chapter 7 exercises 6, 9, 16
• HW #5, due Friday Mar 9:
• Ch 7 exercises 22, 23, 30, 35, 38
• Consider 2x4 matrices with entries from the field ZZ/p, p prime, but call two equivalent if they are related by row operations. How many equivalence classes are there, as a polynomial (!) in p? [Hint: convince yourself that each equivalence class contains exactly one reduced row-echelon matrix.] Answer posted here.
• HW #6, due Friday Mar 16: Comments on the problems and grading here
• Ch 9 exercises 2, 8, 29, 43, 60
• Ch 11 exercises 3, 11
• HW #7, due Friday Mar 23:
• Let G be a product of cyclic groups of prime power orders. Use the Chinese Remainder Theorem to prove that G is a product Z_{m_1} x Z_{m_2} x ... where each m_i divides m_{i+1}. Answer here.
• Ch 11 exercise 37, 39, 47
• Ch 12 exercise 3, 10
• HW #8, due Friday Mar 30:
• Ch 12 exercise 7, 10, 33 [sorry about that, please don't hand it in a second time]
• Ch 13 exercise 6, 10
• Ch 14 exercise 2, 19
• HW #9, due Friday Apr 13:
• Ch 14 exercise 27, 28, 43, 49
• Ch 15 exercise 6, 15, 41
• Prelim #2 is Thursday April 22, same format as the first. Practice problems:
• Ch 6: 23, 26, 41, 55, 56
• Ch 7: 33, 41
• Ch 8: 8
• Ch 9: 15, 22, 31, 37, 48, 53
• Ch 11: 13, 14, 31, 51
• Ch 12: 5, 11, 20, 35, 50
• Ch 13: 3, 4, 5, 7
• Ch 14: 12, 21, 44, 46
• Ch 15: 8, 19
• HW #10, due Friday Apr 27:
• Ch 24 exercise 1, 2, 6, 18
• HW #11, due Wednesday May 9:
• Ch 23: 4, 6, 9, 20

• BONUS SECTION

Some puzzles (and art!) around the Chinese Remainder Theorem by our TA Christine.
Spot It! over F_2: 