Embedded Files
This is Modern Algebra, an undergraduate course.
It is taught at the United States Naval Academy, Annapolis, MD.
We will be using Abstract Algebra: A Geometric Approach by Theodore Shifrin.
Abstract Algebra errata and typographical errors.
Congratulations, Class of 2022 and 2023!
Below is a photograph of Class of 2023 on Friday 26 May 2023.
Instructor: Professor Mee Seong Im
Office: Chauvenet Hall, Office 342, Department of Mathematics, USNA, Annapolis, MD 21402
Office phone number: (410) 293-6776
Email: im [at] usna [dot] edu
Extra Instructions (E.I.): please request for them at least 2 working days in advance (this excludes weekends and Federal Holidays) in order to give Professor Im enough time to do the scheduling. My preference is to meet in-person in my office, rather than over Zoom.
Note: for ALL EIs, please come prepared, with homework problems and proofs attempted in advance and with specific questions in mind.
Mathematics Lab: I will be at the math lab (free tutoring room in Chauvenet Hall, room 130) on Fridays during the 6th period (1430-1520).
All course material will be posted here. Your grades will be regularly posted in Blackboard.
Go to the Google Drive for ALL course material.
Modern Algebra (1 section).
Section 5001 MWF5, CH175, 1330-1420
Section Leader: MIDN Daniel Christie
First Assistant Section Leader: MIDN Caroline Leal
Second Assistant Section Leader: MIDN Hunter Emanus
Breakdown of the points:
Final Exam: 35%
Exams: 45%
Homework: 12%
Group Pop Quizzes (partner list): 08%
Grade Quality Points Point Values
A 4.00 93.5 - 100
A- 3.70 90 - 93.49
B+ 3.30 86.5 - 89.99
B 3.00 83.5 - 86.49
B- 2.70 80 - 83.49
C+ 2.30 76.5 - 79.99
C 2.00 73.5 - 76.49
C- 1.70 70 - 73.49
D+ 1.30 66.5 - 69.99
D 1.00 60 - 66.49
F 0.00 0 - 59.99
Go to the Google Drive for ALL course material.
Academy-wide tutoring (all are free, see USNA Blackboard for the Google Meet links)
Mathematics Lab (via Google Meet CH130), run by the Department of Mathematics faculty members at the US Naval Academy, Annapolis, MD.
Mondays through Fridays, first through sixth period: 0755-1520.
I will be at the math lab on Fridays during the 6th period (1430-1520).
You are allowed to go there as many times as you'd like and talk to the professor in the room about the course material. You are also allowed to go to that room to do your homework or study for your tests.
Academic Center, US Naval Academy, Annapolis, MD. Faculty and Staff (Professional Tutors).
One-on-one professional tutoring during the day (appointments may be necessary).
Evening professional tutoring is available (walk-ins are encouraged).
The Academic Center also offers help via online platforms, like Google Meet, Zoom, etc. in case you cannot leave your room.
Study Groups, a great way to learn from your peers and to teach your peers, and to form friendships. So collaboration is encouraged.
You are ALWAYS welcome to form study groups with your classmates using all the tools that we have (Zoom, Google Meet, Skype, tablets, jamboard, etc.).
Go to the Google Drive for ALL course material.
Keep in mind that for every 1 hour of class, it is recommend that you put in 2 hours of individual or small group study time outside of the class; EIs count toward this.
Friendly reminder: if you get sleepy, feel free to stand during class, get some water, or get some fresh air.
Homework
Your homework assignments are due every Friday at the beginning of the class.
The pages must be stapled and legible. Selected problems will be graded. Check here regularly for updates to your homework assignments since I will regularly add-on more homework problems before or after each lesson. Group work is encouraged but plagiarism will not be tolerated. You should attempt the homework problems on your own first to maximize your understanding of the course material before going to your classmates for hints or help. You are also allowed to show your algebraic or proof techniques to your classmates if you want their feedback on your ideas.
Note that Ted Shifrin does not spoon-feed you, where all you need to do is mimic similar problems. Some are challenging but once you get them, they are fun!! Start on your homework assignments early and please work together with other midshipmen as a team. Feel free to see me as well as email me for hints.
Homework 1 (due Friday 3 Sept): Section 1.1. #1 (compute this without using a calculator but use properties (1) through (8) on page 2), 3, 4, 5, 10, Section 1.2. #1. [Graded 1.1. #3, #5, 1.2. #1.c]
Homework 2 (due Friday 10 Sept): Section 1.2. #2, 3, 4, 5, 6, 7, 8, 9, 11 (these proofs should be 1-2 lines long), 12, 13, 14, Section 1.3, #1. [Graded 1.2. #4, #5, #7, #13, #14]
Homework 3 (due Friday 17 Sept): Section 1.3. #2, 5, 6, 7, 8, 9, 11, 12, 19 (all proofs are at most 2-4 lines), 20.a,b,c,d, 21.a,b,c,d,g, 25, 29, Section 1.4. #1. [Graded 1.3. #5, #6, #20.d, #21.a, 1.4. #1]
Homework 4 (due Friday 24 Sept): Section 1.4. #2, 3, 4, 5, 6, 7, 8, 10, 11, 12, Section 2.1. #3, 8, 9, 10, 11, Section 2.2. #3, 4, 5 (all solutions should take up a maximum of 2-4 lines). [Graded 1.4. #7, 8, 2.1. #3, 2.2. #4]
Homework 5 (due Wednesday 6 Oct): Section 2.2. #6, 7, 10, 12, Section 2.3. #1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 22 (only need to do 2 problems from each section; after doing some easier problems, please try to challenge yourself by doing a slightly more challenging problem so that you have a solid understanding of the concepts).
Homework 6 (due Friday 8 Oct): Section 2.3. #11, 13, 15, 16, 17, 18, 20, 21, 23, Section 2.4. #1.a,b,c, Section 2.5. #1 (only need to do 2 problems from Section 2.3, the 1 problem from Section 2.4, and the 1 problem from Section 2.5).
Homework 7 (due Friday 15 Oct): Section 2.5. #2, 3, 4, 5, 6, 7, 8, 9, 10, 11, Section 3.1. #1, 2, 5 (only need to do 2 problems from Section 2.5 and 2 problems from Section 3.1).
Homework 8 (due Friday 22 Oct): Section 3.1. #6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 23 (only need to do 2 problems from Section 3.1).
Homework 9 (due Friday 29 Oct): Section 3.1. #15, 16, 18, 20, 23, Section 3.2. #1, 2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 15, 16, 17, 18, Section 3.3. #2, 3, 4 (do any 2 problems from Section 3.1, do 2 problems from Section 3.2, and do 2 problems from Section 3.3).
Homework 10 (due Friday 5 Nov): Section 3.3. #5, 6, 7, 8, 9, 10, Section 4.1. #1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (do 2 problems from Section 3.3 and do 2 problems from Section 4.1)
Homework 11 (due Friday 12 Nov): Section 4.1. #16, 17, 18, 19, 20, 21, 22, Section 4.2. #1, 2, 3, 4, 5, 6, 7, 8 (do 2 problems from Section 4.1 and do 2 problems from Section 4.2)
Homework 12 (due Friday 19 Nov): Section 4.2. #9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, Section 5.1. #1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (do 2 problems from Section 4.2; do 2 problems from Section 5.1)
Homework 13 (due Friday 3 Dec): Section 5.1. #11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, Section 5.3. #1, 2, 3, 4, 5, 6, 7, 8 (do 2 problems from Section 5.1; do 2 problems from Section 5.3)
Homework 14 (due Friday 10 Dec): Section 6.1. #1 (only do 4), #2-25, Section 6.2. #1-17, Section 6.3. #1-35 (do any 2 problems from Section 6.1; do any 2 problems from Section 6.2; do any 2 problems from Section 6.3)
In mathematics, the quaternion number system extends the complex numbers. Multiplication of quaternions is noncommutative. That is, quaternions Q = {±1, ±i, ±j, ±k : i^2 = j^2 = k^2 = -1, ij = k, jk = i, ki = j, ji = - k, kj = -i, ik = -j} form a nonabelian group. Notice that ijk = -1.
An application of binary numbers and integers (enjoy!):
Go to the Google Drive for ALL course material.
Course Schedule
Lesson 1 (Mon 23 Aug): The Integers: Integers, Mathematical Induction, and the Binomial Theorem (Chapter 1.1)
Properties of integers, Principle of Mathematical Induction, Principle of Complete/Strong Induction.
Lesson 2 (Wed 25 Aug): The Integers: Integers, Mathematical Induction, and the Binomial Theorem (Chapter 1.1)
Practice proving truth statements (equalities and inequalities) using (strong) induction, prove the Binomial Theorem using induction.
Lesson 3 (Fri 27 Aug): The Integers: The Euclidean Algorithm, Prime Numbers, and Factorization (Chapter 1.2)
Divisors, Division Algorithm, greatest common divisor (gcd).
Lesson 4 (Mon 30 Aug): The Integers: The Euclidean Algorithm, Prime Numbers, and Factorization (Chapter 1.2)
Euclidean algorithm (if d = gcd(a,b), then find integers m and n such that d = ma + nb), prime and composite numbers, relatively prime, properties of prime numbers.
Lesson 5 (Wed 1 Sept): The Integers: The Euclidean Algorithm, Prime Numbers, and Factorization (Chapter 1.2)
Properties of prime numbers, Fundamental Theorem of Arithmetic, prove the existence of infinitely many prime numbers.
Lesson 6 (Fri 3 Sept): The Integers: Modular Arithmetic and Solving Congruences (Chapter 1.3), Pop Quiz 1
Modulo (a is congruent to b mod m), properties of modular arithmetic, Divisibility Criteria in base 10 and examples and proofs.
Lesson 7 (Tues 7 Sept): The Integers: Modular Arithmetic and Solving Congruences (Chapter 1.3)
Recall Divisibility Criteria, discuss that any perfect square is congruent to either 0 or 1 mod 4, Fermat's proposition (given prime p and any integer n, n^p is congruent to n mod p), prove that there are infinitely-many primes of the form 4k+3, where k is a natural number, discuss and prove how to solve for x if gcd(c,m)=1 and cx is congruent to b mod m.
Lesson 8 (Wed 8 Sept): The Integers: Modular Arithmetic and Solving Congruences (Chapter 1.3)
Do examples on how to solve for x if gcd(c,m)=1 and cx is congruent to b mod m, solve for x if cx is congruent to b mod m (general case), Chinese Remainder Theorem, Improved Chinese Remainder Theorem, and examples.
Lesson 9 (Fri 10 Sept): The Integers: Z_m, Rings, Integral Domains, and Fields (Chapter 1.4), Pop Quiz 2
Prove Improved Chinese Remainder Theorem and do an example, addition, multiplication, and properties for Z_m, rings, commutative rings, zero-divisors, integral domain, fields.
Lesson 10 (Mon 13 Sept): The Integers: Z_m, Rings, Integral Domains, and Fields and From the Integers to the Complex Numbers: The Rational Numbers (Chapters 1.4 and 2.1)
Finite rings and integral domains, multiplicative inverses in a finite field Z_p, noncommutative rings, rational numbers as equivalence classes of ordered pairs, field of fractions for integral domains (cool!! Do you think they have a geometric meaning?), ordering (trichotomy and closed under + and *), ordered fields.
Lesson 11 (Wed 15 Sept): From the Integers to the Complex Numbers: The Rational Numbers and Real Numbers (Chapters 2.1 and 2.2)
Ordered fields, rational numbers form an ordered field, there is a rational number between any two rational numbers, constructing infinitely-many rational numbers between (a/b)^2 and 2, bounded above, upper bound of a subset S, least upper bound.
Lesson 12 (Fri 17 Sept): From the Integers to the Complex Numbers: From the Rational Numbers to the Real Numbers (Chapter 2.2), Pop Quiz 3
Least upper bound, least upper bound property, real numbers, irrational numbers, there is a rational number between any two real numbers, constructing Q[sqrt{2}], which is a field containing Q and is contained in R.
Lesson 13 (Mon 20 Sept): From the Integers to the Complex Numbers: The Complex Numbers (Chapter 2.3)
Construct a field strictly between Q and R (for example, Q[sqrt(2)]), define i such that i^2 = -1, prove that C is a field, represent C by ordered pairs of real numbers in order to view C as the complex plane, geometry associated to C, Re(z), Im(z), complex conjugate, modulus (absolute value), polar form, how to multiply two complex numbers, deMoivre's Theorem: given any integer n and z = r(cos theta + i sin theta), z^n = r^n (cos (n theta) + i sin (n theta)).
Lesson 14 (Wed 22 Sept): From the Integers to the Complex Numbers: The Complex Numbers (Chapter 2.3)
Recall deMoivre's Theorem, find 1/z given a nonzero complex number z, |z|^2 = z\bar{z}, find all the nth roots of unity, understand the (distinct) nth roots of unity geometrically, find all the nth roots of a = r(cos theta + i sin theta), find a^{1/3} if a = -2 + 2i.
Lesson 15 (Fri 24 Sept): From the Integers to the Complex Numbers: The Complex Numbers (Chapter 2.3), Pop Quiz 4
Find the three complex solutions to z^3 = a = -2 + 2i, understand where these roots are located geometrically on the complex plane, investigate the correspondence between trigonometric and exponential functions using complex numbers to derive e^{iθ} = cos θ + i sin θ, so z = r e^{iθ}.
Lesson 16 (Mon 27 Sept): From the Integers to the Complex Numbers: The Quadratic and Cubic Formulas (Chapter 2.4)
Solve a quadratic polynomial over complex numbers, solve a cubic polynomial over the complex numbers, find the 3 roots of z^3 - 3z + 1 = 0, which are the following: let ν = ξ = e^{2π i /9}, νω = ξ^4, νω^2 = ξ^7, where ω = e^{2π i/3}. Then z = ξ + ξ^{-1}, z = ξ^4 + ξ^{-4}, and z = ξ^7 + ξ^{-7}.
Lesson 17 (Wed 29 Sept): From the Integers to the Complex Numbers: The Isometries of R and C (Chapter 2.5)
Review finding the solutions to the cubic equation z^3 - 3z + 1 = 0, definition of isometry for real numbers, classify all isometries f over R such that f(0)=0, classify all isometries over R, fixed points.
Optional (Thurs 30 Sept): Walk-in Extra Instruction (Walk-in EIs)
Walk-in EIs during 1st and 2nd periods: 0755-0945. I'll be in my office in CH342 if you want to drop by. If there are many people, we will go to the conference room 5 doors to the left of my office.
Lesson 18 (Fri 1 Oct): Midshipmen-Driven Review: go through class notes and past and current homework assignments with your classmates, Pop Quiz 5
There will be a substitute today. Professor Vrej Zarikian will be your instructor.
Professor Linda Shivok will be in the Math Lab today during the 6th period (1430 -1520) in CH130.
How to study for your test? Study what we discussed in class and review your homework sets.
Optional (Sun 3 Oct): Extra Instruction (EI)
Please let me know if you would like to meet and all the hours you are available. Then I will post the hours when we will meet in CH342 or in a conference room on the third deck of Chauvenet Hall. Let me know if you plan to drop by-- if the weather is nice, we may meet outside instead.
Lesson 19 (Mon 4 Oct): From the Integers to the Complex Numbers: The Isometries of R and C (Chapter 2.5)
Definition of isometry for complex numbers, f maps a circle of radius r centered at a to a circle of radius r centered at f(a) if f is an isometry over C, classify all isometries f over C such that f(0)=0 and f(1)=1, classify all isometries over C, fixed points.
Optional (Mon 4 Oct): Extra Instruction (EI)
Please let me know if you would like to meet and all the hours you are available. Then I will post the hours when we will meet in CH342 or in a conference room on the third deck of Chauvenet Hall. Let me know if you plan to drop by-- if the weather is nice, we may meet outside instead.
Optional (Tues 5 Oct): Walk-in Extra Instruction (EI)
Please let me know if you would like to meet and all the hours you are available. Then I will post the hours when we will meet in CH342 or in a conference room on the third deck of Chauvenet Hall. Let me know if you plan to drop by-- if the weather is nice, we may meet outside instead.
EI: 1300-1530, CH342 (my office), meet in CH320 (seminar room), I will meet with a few midshipmen so you are welcome to drop-in if you'd like.
EI: 1530-1700, CH342 (my office), meet in CH351 (seminar room), I will meet with a midshipman so you are welcome to drop-in if you'd like.
Lesson 20 (Wed 6 Oct): Test 1
Test 1 will cover Chapter 1 through Chapter 2.4. The cubic formula will not be on the test.
There are 7 problems, and some problems have multiple parts.
Lesson 21 (Fri 8 Oct): Polynomials: The Euclidean Algorithm (Chapter 3.1)
Review isometries of the real line and the complex plane. Polynomials of degree n over a ring R, degree of a polynomial, monomials, coefficients, leading coefficient, monic polynomials, how to add and multiply two polynomials whose coefficients are in a ring R, R[x] is a commutative ring if R is a commutative ring, deg(f(x) g(x)) = deg(f) + deg(g) if f(x) ≠ 0 and g(x) ≠ 0 in R[x] where R is an integral domain; furthermore, R[x] is an integral domain if R is integral domain.
Lesson 22 (Wed 13 Oct): Polynomials: The Euclidean Algorithm (Chapter 3.1), Pop Quiz 6
Prove the division algorithm for polynomials in F[x] where F is a field: given two nonzero polynomials f(x) and g(x) in F[x], there exist unique polynomials q(x) and r(x) in F[x] such that f(x) = q(x)g(x) + r(x), where deg(r(x)) < deg(g(x)) or r(x) = 0. Do an example.
Lesson 23 (Fri 15 Oct): Polynomials: The Euclidean Algorithm (Chapter 3.1)
Examples of the division algorithm for polynomials in F[x], where F is a field, the Remainder Theorem (given a polynomial f(x) in F[x], x-c will divide into f(x) with remainder f(c)), example, root of a polynomial, Root-Factor Theorem (given f(x) in F[x], x-c is a factor of f(x) if and only if c is a root of f(x)), example. Let R = Z_6 and multiply the polynomials f(x) = 2x+4 and g(x) = 3x+3 ∈ Z_6[x] (you can see that bizarre things can occur when R has zero-divisors). Definition of irreducible polynomials, Euclidean algorithm for polynomials, greatest common divisor (gcd) d(x) for polynomials f(x) and g(x).
Bring completed Pop Quiz 6 to class today. Remember, external resources are not allowed on Quiz 6, except your partner.
Professor Daphne Skipper will be in the Math Lab today during the 6th period (1430-1520) in CH130.
Lesson 24 (Mon 18 Oct): Polynomials: Roots of Polynomials (Chapter 3.2)
Euclidean algorithm: find the monic polynomial d(x) = gcd(f(x), g(x)) of f(x) = x^3-8 and g(x) = x^2-x-2 in Q[x] and also find s(x) and t(x) such that d(x) = s(x)f(x) + t(x)g(x). Gavin finds the gcd for f(x) = x^3-1 and g(x) = x^3-x^2-x+1 ∈ Z_3[x]. Polynomials are relatively prime if gcd(f(x), g(x)) = 1. If f(x) ∈ F[x] is irreducible and f(x)|g(x)h(x), then f(x)|g(x) or f(x)|h(x). Unique factorization in F[x]. The degree 2 polynomial f(x) = x^2 - 1 ∈ Z_8[x] has 4 roots ±1 and ±3, so f(x) has two different factorizations: f(x) = (x-1)(x+1) = (x-3)(x+3); therefore, polynomial factorization is not unique for rings with zero-divisors (it seems that rings with zero-divisors are more difficult to study, yet possibly more interesting?).
Fundamental Theorem of Algebra: given a polynomial f(x) in C[x] of degree ≥ 1, f(x) has a root in C. Find the smallest possible field K containing all the roots of a polynomial f(x) ∈ Q[x]. Prove that if K is a field contained in C, then K contains Q. If K is a field contained in R, then K contains Q. If F and K are fields such that K⊇F and α ∈ K, then F[α] = {p(α)∈ K : p(x) ∈ F[x]}. If α, β ∈ K, then F[α, β] = (F[α])[β].
Lesson 25 (Wed 20 Oct): Polynomials: Roots of Polynomials (Chapter 3.2)
A thorough investigation of Q[√2], compare Q[√3 i] and Q[√3, i], compare Q[√3+i] and Q[√3, i]. Prove that given f(x) ∈ F[x], where K ⊇ F is a field containing a root α of f(x), then F[α] ⊆ K is a field. Find the multiplicative inverse of β = α^2 + α - 1, where α satisfies α^3 + α + 1 =0.
We may have Trident Days visitors today.
Lesson 26 (Fri 22 Oct): Polynomials: Roots of Polynomials and Polynomials with Integer Coefficients (Chapters 3.2-3.3), Pop Quiz 7
Daniel will go through the division algorithm and Euclidean algorithm with us for the last example from Wednesday's class to find polynomials s(x) and t(x) such that 1 = s(x) p(x) + t(x) g(x), where p(x) = x^2 + x - 1 and g(x) = x^3 + x + 1. Discuss the finite field Z_2[α] that satisfies α^2+α+1=0 (here, α and α+1 are multiplicative inverses). Field extensions. f(x) in Q[x] splits in K if f(x) can be written as a product of linear polynomials in K[x]. If f(x) splits in K but not in any field E such that F ⊊ E ⊊ K, then K is called a splitting field of f(x). Examples of how to construct splitting fields. Rational roots theorem. If f(x) is a monic polynomial of degree n ≥ 1, any rational root must be an integer r such that r divides the constant term of f. Examples of how to find rational roots of polynomials.
Your partner for Pop Quiz 7 is assigned. External resources are not allowed on Quiz 7, except your partner.
Lesson 27 (Mon 25 Oct): Polynomials: Polynomials with Integer Coefficients (Chapter 3.3)
Review Rational Roots Theorem. Gauss' Lemma, Method of Undetermined Coefficients. Given f(x)=a_n x^n + ... + a_1 x + a_0 ∈ Z[x], if for some prime p, a_n is not congruent to 0 mod p and \bar{f}(x) in Z_p[x] is irreducible, then f(x) is irreducible in Q[x].
Bring completed Pop Quiz 7 to class today.
Lesson 28 (Wed 27 Oct): Polynomials: Polynomials with Integer Coefficients and Homomorphisms and Quotient Rings: Ring Homomorphisms and Ideals (Chapter 3.3 and 4.1)
Eisenstein's Criterion and examples. Given two rings R and S, ϕ: R → S is a ring homomorphism if for any a, b ∈ R, ϕ(a+b) = ϕ(a) + ϕ(b), ϕ(ab) = ϕ(a)ϕ(b), and ϕ(1_R) = 1_S. Prove that given a ring homomorphism ϕ: R → S, ϕ(0_R) = 0_S and the image of ϕ, which is denoted by im(ϕ) = ϕ(R), is a subring of S. Go through many examples of ring homomorphisms.
Lesson 29 (Fri 29 Oct): Homomorphisms and Quotient Rings: Ring Homomorphisms and Ideals (Chapter 4.1), Pop Quiz 8
Recall (ring) homomorphisms and prove that ϕ:C→C, where ϕ(z) = \bar{z} is a ring homomorphism. Given a commutative ring R, the ideal generated by a is of the form ⟨a⟩ = {ra: r ∈ R}. Principal ideal and principal ideal domain (PID). The set of integers and F[x] are PIDs. If I ⊊ R is an ideal and a prime number p ∈ I, then I = ⟨p⟩. Similarly, if I ⊊ F[x] is an ideal, f(x) ∈ I and f(x) is irreducible in F[x], then I = ⟨f(x)⟩. If f(x) ∈ F[x] is irreducible in F[x] and K ⊇ F is a field containing a root α of f(x), then the ideal of all polynomials in F[x] vanishing at α is generated by f(x).
Lesson 30 (Mon 1 Nov): Homomorphisms and Quotient Rings: Ring Homomorphisms and Ideals (Chapter 4.1) (Chapter 4.2)
Definition of kernel of a ring homomorphism, an ideal, example of an integral domain that is not a PID: consider Z[x] and the ideal I = ⟨3, x+2⟩, equivalent class of a (mod I), R/I = set of equivalent classes, Z_m = Z/⟨m⟩, quotient rings. Given a commutative ring R and a proper ideal I of R, R/I is also a commutative ring.
Lesson 31 (Wed 3 Nov): Homomorphisms and Quotient Rings: Ring Homomorphisms and Ideals and Isomorphisms and the Fundamental Homomorphism Theorem (Chapters 4.1 and 4.2)
Review quotient rings R/I. We write the equivalent classes (elements) in R/I as bar{a} or a (mod I). Examples of quotient rings: Z_6/⟨2⟩, Q[x]/⟨x^2-2⟩ ≅ Q[√2], Z_2[x]/⟨x^2+x+1⟩ ≅ {0, 1, x, x+1}, where the last quotient ring is a field with 4 elements. Given a ring homomorphism ϕ: R→S, ϕ is an isomorphism if it is 1-1 and onto. If ϕ is an isomorphism, we say R is isomorphic to S, and write R ≅ S. A ring homomorphism ϕ: R→S is 1-1 ⇔ ker ϕ = ⟨0⟩. Show that ϕ: C→C, where ϕ(z) = bar{z}, is an isomorphism. Show that ϕ: R[x]/⟨x^2+1⟩ → C, where ϕ(f(x) mod ⟨x^2+1⟩) = f(i), is a well-defined isomorphism.
Lesson 32 (Fri 5 Nov): Homomorphisms and Quotient Rings: Isomorphisms and the Fundamental Homomorphism Theorem (Chapter 4.2), Pop Quiz 9
Given two rings R and S, we construct a new ring called the direct product R x S of R and S (0 := (0,0) and 1:= (1,1), and addition and multiplication are done component-wise, so R x S is a ring). Given an isomorphism ϕ: R→S, if a ∈ R is a unit, then ϕ(a) ∈ S is also a unit. Given an isomorphism ϕ: R→S, if a ∈ R is a zero-divisor, then ϕ(a) ∈ S is also a zero-divisor. Is Z_4 ≅ Z_2 x Z_2? No, because Z_4 has 1 zero divisor while Z_2 x Z_2 has 2 zero divisors. Fundamental Homomorphism Theorem: given commutative rings R and S and a ring homomorphism ϕ: R→S, we have R/ker ϕ ≅ S.
LCDR Jeff Lineberry will be in the Math Lab today during the 6th period (1430 -1520) in CH130.
Lesson 33 (Mon 8 Nov): Homomorphisms and Quotient Rings: Isomorphisms and the Fundamental Homomorphism Theorem (Chapter 4.2)
Review Fundamental Homomorphism Theorem. Show that given the evaluation ring homomorphism ev_i: R[x]→C, where ev_i(f(x)) = f(i), R[x]/⟨x^2+1⟩ ≅ C. Show that Z_{12}/⟨3⟩ ≅ Z_3. Show that Z[x]/⟨2x-1⟩ ≅ Z[1/2]. Note that Q[x]/⟨x^2⟩ is not an integral domain since x is a zero-divisor. Given a field F and a polynomial f(x) ∈ F[x], F[x]/⟨f(x)⟩ is a field ⇔ f(x) is irreducible in F[x].
Lesson 34 (Wed 10 Nov): Homomorphisms and Quotient Rings: Isomorphisms and the Fundamental Homomorphism Theorem (Chapter 4.2)
Given the ring Z_3[x], the polynomial f(x) = x^3-x^2+1 is irreducible. Gavin and Molly will find the multiplicative inverse of x^2+x+1 ∈ Z_3[x]/⟨f(x)⟩ by using the Euclidean algorithm (they saw that 1 = -(x+1)(x^3-x^2+1) + (x^2-x-1)(x^2+x+1), which imply that x^2-x-1 is the multiplicative inverse of x^2+x+1 in the field Z_3[x]/⟨f(x)⟩). Paul will finish the section by proving if f(x) ∈ F[x] is an irreducible polynomial, then K = F[x]/⟨f(x)⟩ contains a root α of f(x) and K ≅ F[α]. Paul will also prove that any polynomial has a splitting field. Finally, Paul will discuss multiple examples, including: given a ∈ F and f(x) = x^2 - a is irreducible, then K = F[x]/⟨f(x)⟩ is the splitting field of f(x) because K = F[x]/⟨f(x)⟩ ≅ F[√a]. As a second example, given f(x) = x^3-2 ∈ Q[x], K = Q[x]/⟨x^3-2⟩ ≅ Q[2^{1/3}]; however, K is not the splitting field of f(x) but instead, L = K[y]/⟨y^2+αy+α^2⟩ ≅ K[2^{1/3}, √3i] is the splitting field of f(x).
Lesson 35 (Fri 12 Nov): Field Extensions: Vector Spaces and Dimension (Chapter 5.1), Pop Quiz 10 [on top of having a partner, you may phone-a-friend also; I will explain this in class. Also, the email I sent you contains more details.]
Paul discussed how to find the splitting field for the irreducible polynomial x^3+x+1 ∈ Z_2[x]. Let α be a root of x^3+x+1. Then Z_2[x]/⟨x^3+x+1⟩ ≅ Z_2[α]. In the field extension, he showed that the other two roots are α^2 + α and α^2; so Z_2[α] is the splitting field for the irreducible polynomial. Definition of vector spaces, subspaces, linear combination. Span(v_1,..., v_k) is a subspace of V, called the subspace spanned by v_1,..., v_k. Definition of linearly independent and linearly dependent.
Lesson 36 (Mon 15 Nov): Field Extensions: Vector Spaces and Dimension (Chapter 5.1)
Definition of finite-dimensional vector space, definition of dimension of a vector space V, dim {0} = 0. If V is a finite-dimensional vector space and V = span(v_1,...,v_k), then there exists a subset of {v_1,...,v_k} such that the subset forms a basis for V. If V is a finite-dimensional vector space and v_1,...,v_k span V and w_1,...,w_l ∈ V are linearly independent, then l ≤ k. Any two bases for a finite-dimensional vector space V have the same number of elements. Given fields K and F satisfying F ⊆ K, if K is a finite-dimensional vector space over F, then K is a field extension of F of degree dim_F(K) = deg_F(K) = [K:F].
Lesson 37 (Wed 17 Nov): Field Extensions: An Introduction to Finite Fields (Chapter 5.3), Pop Quiz 11 [on top of having a partner, you may phone-a-friend also]
If K is a field extension of F and α ∈ K is a root of an irreducible polynomial f(x) ∈ F[x] of degree n, then [F[α]:F] = n = deg_F(f(x)). If [K:F] = n and α ∈ K is a root of an irreducible polynomial f(x) ∈ F[x], then deg_F(f(x))|n.
A field F has characteristic m if m is the smallest value such that m*1_F = 1_F + ... + 1_F = 0. A field has characteristic 0 if no such m exists. If F is a field, then char(F) = 0 or p, where p is prime. If F is a field with char(F) = p, then F contains a copy of Z_p; so F is a vector space over Z_p. If F is a vector space over Z_p of dimension n ∈ N, then F has p^n elements for some n ∈ N. Given a prime p and n ∈ N, there exists a field F_q with q = p^n elements. Let q = p^n and q' = p^{n'}. Then F_q ⊆ F_{q'} if and only if n|n'.
Lesson 38 (Fri 19 Nov): Midshipmen-Driven Review
Midshipmen-driven class (bring specific questions or problems you want us to solve as a class).
LCDR Jeff Lineberry will be in the Math Lab today during the 6th period (1430 -1520) in CH130. If you end up going to the Math Lab for some help, please be sure to thank LCDR Lineberry for his time.
Lesson 39 (Mon 22 Nov): Test 2
LCDR Chris Smith will be your instructor during the 5th period. Please be nice and good luck everyone!
Test 2 will cover Chapter 2.5 to Chapter 5.1. There are 8 problems, with some problems having multiple parts, spanning over 10 pages.
You must know the definitions and big results (Theorems, Propositions, Lemmas, Corollaries) from each section. Do not memorize the proofs to these results, but know how to use the definitions and apply the big results (you should know how to do the basic problems from the 7 sections listed below). If you are now comfortable with the examples in the book, you may ignore them.
The following will be on your test:
Section 2.5 (pages 73-77): definition of isometry of R and C, fixed points, proper isometries and improper isometries
Section 3.1 (pages 83-89): everything but the content below
Section 3.2 (pages 95-100, omit the proof on page 96): everything but the content below
Section 3.3 (pages 105-110): everything
Section 4.1 (pages 114-121): everything
Section 4.2 (pages 125-131): everything but the content below
Section 4.3: N/A
Section 5.1 (pages 150-155): everything
Section 5.2: N/A
The following will not be on your test:
Section 2.5: omit everything after glide reflection (page 78)
Section 3.1: omit everything after rational functions, Partial Fractions Decomposition (Theorem 1.9, page 89)
Section 3.2: omit everything after Descartes' Rule of Signs (Theorem 2.3, page 100)
Section 3.3: N/A
Section 4.1: N/A
Section 4.2: omit everything after transcendental and algebraic numbers (page 132)
Section 4.3: omit everything
Section 5.1: N/A
Section 5.2: omit everything
Have a safe Thanksgiving Break, everyone!
Lesson 40 (Mon 29 Nov): Groups: The Basic Definitions (Chapter 6.1)
Review with examples: Given a prime p and n ∈ N, there exists a field F_q with q = p^n elements. Let q = p^n and q' = p^{n'}. Then F_q ⊆ F_{q'} if and only if n|n'. Discuss how to explicitly find a field of order p^n.
Definition of symmetry. What are possible symmetries of an equilateral triangle? Definition of a group and abelian group. Abelian groups include integers, Z_m, Q, R, C, under addition. Abelian multiplicative groups include the units R^* = R - {0} in R, C^*, F^*, and the Klein four group. Nonabelian groups include symmetries of an equilateral triangle, the quaternion group, the general linear group GL_2(R), and the special linear group SL_2(R). Prove some fundamental properties about groups: if G is a group with identity element e, then:
ea = ae = a and e'a = ae' = a ⇒ e = e'.
If ab = e, then b = a^{-1}. As a consequence, each element has a unique inverse.
If a, b, c ∈ G and ac = bc (or ca = cb), then a = b.
If a ∈ G, then (a^{-1})^{-1} = a.
If a, b ∈ G, then (ab)^{-1} = b^{-1}a^{-1}.
Lesson 41 (Wed 1 Dec): Groups: The Basic Definitions and Group Homomorphisms and Isomorphisms (Chapters 6.1 - 6.2)
Prove that the units R^* in a ring R form a group with respect to multiplication operation in R. H is a subgroup if H is not empty and it is closed under multiplication and inverses (if a, b ∈ H, then ab^{-1} ∈ H). H is a proper subgroup of a group G if H ≠ {e} or H ≠ G. A cyclic subgroup ⟨a⟩ = {a^n: n ∈ Z} is generated by a. If ⟨a⟩ = G, then G is cyclic. If |⟨a⟩| < ∞, then the order of the element a is the smallest n ∈ N such that a^n = e. G is finite if |G| < ∞. The number of elements of G is called the order of G, and is written |G|. Note that the order(a) = order(⟨a⟩). All subgroups of the integers are cyclic ⟨m⟩. Find all cyclic subgroups of Z_{12}. Show that (Z_4, +) is not isomorphic to the Klein 4-group (V, *) by looking at the order of the elements. Considering the symmetries of an equilateral triangle of order 3! = 6, the identity symmetry has order 1, rotations R_1 (120 CCW) and R_2 (240 CCW) have order 3, and reflections F_1, F_2, F_3 have order 2.
Let G, G' be groups. ϕ : G → G' is a group homomorphism if ϕ(ab) = ϕ(a)ϕ(b) for all a, b ∈ G. If ϕ : G → G' is a (group) homomorphism, then ϕ(e) = e', where e ∈ G and e' ∈ G' are identity elements, and (ϕ(a))^{-1} = ϕ(a^{-1}) for all a ∈ G. The kernel of a group homomorphism ϕ : G → G' is defined to be ker ϕ = {a ∈ G : ϕ(a) = e'}. Prove that ker ϕ is a subgroup of G, i.e., ker ϕ ≤ G, and ker ϕ = {e} ⇔ ϕ is 1-1. A group homomorphism ϕ : G → G' is an isomorphism if and only if it is 1-1 and onto G', and we write G ≅ G'. Assuming that ϕ : G → G' is an isomorphism,
G is a finite group ⇒ G' is a finite group, and |G| = |G'|,
For all a ∈ G, order(a) = order(ϕ(a)),
If G is is abelian, then G' is abelian.
Lesson 42 (Fri 3 Dec): Groups: Group Homomorphisms and Isomorphisms (Chapters 6.2), Pop Quiz 12
Definition of a permutation of a set A, which is a function π : A → A that is bijective. Prove that Perm(A) is a group with respect to the operation of composition of functions. If A = {1, 2, 3, ..., n}, then S_n := Perm(A), the symmetric group on n letters. The cardinality of S_n is |S_n| = n!. Example: write the symmetries of an equilateral triangle using a permutation notation, using two rows.
An example of group homomorphism is ϕ : C^* → S^1 = {w ∈ C: |w| = 1}, where ϕ(z) = z/|z|. Show that ϕ is a group homomorphism and that ϕ is surjective. Find the kernel of ϕ = { z = x+ iy ∈ C^* : y = 0 and Re(z) > 0 }. Another example of a group homomorphism is the following: G is a cyclic group satisfying |G| = n ⇒ G = ⟨a⟩ = {a^i : i > 0}, with a^n = e, a^k ≠ e for 0 < k < n. Let ϕ: Z_n → ⟨a⟩, where ϕ(k) = a^k. Show that ϕ is well-defined, and then ϕ is a group homomorphism. Then show that ϕ is 1-1 and onto. This imply that whenever we have a cyclic group G of order n, then G = ⟨a⟩ ≅ Z_n, and it is abelian.
Professor Im will open SOF (opened on Thursday 2 Dec 2021).
Lesson 43 (Mon 6 Dec): Groups: Cosets, Normal Subgroups, and Quotient Groups (Chapter 6.3), Pop Quiz 13
Definition of (left) coset of a subgroup H ≤ G: aH = {ah: h ∈ H}. If G is abelian, then the group operation is + and we write left cosets as a + H = {a + h: h ∈ H}. An example: G = Z and H = 2Z. Then there are two left cosets 0 + H (even integers) and 1+ H (odd integers). Another example: G = symmetries of an equilateral triangle and H = {1, F_1}. Then there are three cosets:
1 H = {1, F_1} = F_1 H = {F_1, F_1 F_1 = 1},
R_1 H = {R_1, R_1 F_1 = F_3} = F_3 H = {F_3, F_3 F_1 = R_1}, and
R_2 H = {R_2, R_2 F_1 = F_2} = F_2 H = {F_2, F_2 F_1 = R_2}.
Example: Let G = R^2, which is a group under addition (R^2, +). Let H ≤ G, where H = { (x,y)∈R^2 : y = mx }, a 1-dimensional vector subspace, so H is also a subgroup. Then b + H are cosets of H, where b = (b_1,b_2) ∈ R^2, and they are called affine subspaces.
Two cosets have nonempty interaction if and only if they are identical. G = a disjoint union of (left) cosets. Definition of the index of H in G: [G:H] = # of distinct cosets of H in G. Example: G = Z and H = nZ, then [G:H]=n, where the cosets are H, 1 + H, 2 + H, ..., (n-1) + H. Another example: G = symmetries of equilateral triangle, H = {1, F_1}. Then [G:H] = 3.
Theorem [Lagrange]. Assume G is a finite group and H ≤ G, a subgroup of G. Then [G:H] = |G|/|H|. If H ≤ G and |G| < ∞, then |H|||G|. If H ≤ G, |G| < ∞, and if |H| > (1/2)|G|, then |H|=|G|, and H = G. If G is a finite group and a is an arbitrary element in G, then order(a)||G|. If |G|= p and p is prime, then G is cyclic. If |G| = n and a ∈ G is arbitrary, then a^n = e. H ≤ G is a normal subgroup of G, and write H ⊴ G, if aHa^{-1} ⊆ H for all a ∈ G. The product aha^{-1} of elements of G is called the conjugate of h by a. We also say g and g' are conjugate if aga^{-1} = g' for some a ∈ G. If H ⊴ G, then G/H is a group with identity eH = H and multiplication defined by (aH)(bH) = (ab)H.
Lesson 44 (Wed 8 Dec): Groups: Cosets, Normal Subgroups, and Quotient Groups (Chapter 6.3) and Midshipmen-Driven Review
If H ⊴ G, then G/H is a group with identity eH = H and multiplication defined by (aH)(bH) = (ab)H (finish proving this). Remark: [G : H] < ∞, then |G/H| = [G : H]. When |G| < ∞, then |G/H| = |G|/|H|.
Example: when G is abelian, then H ≤ G is also abelian ⇒ H ⊴ G, and G/H is an abelian subgroup.
Example: G = GL_n(F) and H = SL_n(F) ⊴ GL_n(F). This is because if det(B) = 1 and A ∈ GL_n(F), then det(ABA^{-1}) = det(B) = 1. So ABA^{-1} ∈ SL_n(F).
Fundamental Homomorphism Theorem: if ϕ : G ↠ G' is a (group) homomorphism onto G', then G/ker ϕ ≅ G'. If ϕ : G → G' is a (group) homomorphism, then im ϕ := {y ∈ G' : ϕ(a) = y for some a ∈ G}. Corollary: im ϕ ≤ G'. Corollary: G/ker ϕ ≅ im ϕ.
Lesson 45 (Fri 10 Dec): Midshipmen-Driven review. Last Day with Professor Im. Woo hoo!! =o)
Fill out and submit an SOF.
Optional (Sun 19 Dec): Walk-in Extra Instruction with MIDN Paul Zimmer (EI)
MIDN Paul Zimmer volunteered to run EIs during the week of the Final Exam. Once one person requests for an EI, I will post the date, time, and the room number here. Please refresh this page regularly. Everyone is welcome!!
Optional (TBD): Walk-in Extra Instruction with MIDN Paul Zimmer (EI)
MIDN Paul Zimmer volunteered to run EIs during the week of the Final Exam. Once one person requests for an EI, I will post the date, time, and the room number here. Please refresh this page regularly. Everyone is welcome!!
Optional (TBD): Walk-in Extra Instruction (EI)
Although classes officially end on Friday 10 December, I will be available for EIs after 10 December. You must bring at least 1 other midshipman in my SM362 class to the EI, for a combined total of at least 2 midshipmen. So please check with them for their schedule. Once we schedule an EI, I will post the date, time, and room number here so that other midshipmen could attend and participate if they want to. So please refresh this page regularly.
Optional (TBD): Walk-in Extra Instruction (EI)
Although classes officially end on Friday 10 December, I will be available for EIs after 10 December. You must bring at least 1 other midshipman in my SM362 class to the EI, for a combined total of at least 2 midshipmen. So please check with them for their schedule. Once we schedule an EI, I will post the date, time, and room number here so that other midshipmen could attend and participate if they want to. So please refresh this page regularly.
Optional (TBD): Walk-in Extra Instruction (EI)
Although classes officially end on Friday 10 December, I will be available for EIs after 10 December. You must bring at least 1 other midshipman in my SM362 class to the EI, for a combined total of 2 midshipmen. So please check with them for their schedule. Once we schedule an EI, I will post the date, time, and room number here so that other midshipmen could attend and participate if they want to. So please refresh this page regularly.
Final Exam (Mon 20 Dec, 1300-1600, CH157): The final exam is cumulative.
There are 20 problems.
For Problem 18b, use the rectangular paper, in the exam packet, whose vertices have been labeled with 1, 2, 3, 4.
The following will be on your final:
Study definitions, theorems, lemmas, and big ideas from each section. Review your quizzes, tests, and homework problems.
All the material for Test 1
All the material for Test 2
Section 5.3 (pages 165-167): everything
Section 6.1 (pages 171-176): everything
Section 6.2 (pages 181-185): everything
Section 6.3 (pages 188-195): everything but the content below
The following will not be on your final:
Section 5.3: N/A
Section 6.1: N/A
Section 6.2: N/A
Section 6.3: omit Theorem 3.11: The set of translations forms a normal subgroup H of the group G of isometries of C (page 195)
Your Final Exam and the solutions are posted in the Google Drive.
Have a wonderful Winter Break!!
Go to the Google Drive for ALL course material.
Go to the Google Drive for ALL course material.
If you are enjoying this course material, a continuation of Abstract Algebra is:
Abstract Algebra 2,
This course includes more advanced abstract algebraic structures and concepts, such as groups, symmetry, group actions, counting principles, symmetry groups of the regular polyhedra, Burnside's Theorem, isometries of the 3-dimensional space R^3, Galois Theory, and affine and projective geometry.
Also check out the following link Modern Algebra II.
Here is an application of modular Galois theory to algebraic geometry (resolutions of singularities).
Algebraic Number Theory (see, for example, Algebraic Number Theory by J.S. Milne),
Algebra (see, for example, Algebra by Thomas W. Hungerford),
Homological Algebra.
See, for example, An Introduction to Homological Algebra by Charles Weibel.
Also see, for example, An Introduction to Homological Algebra by Joseph J. Rotman.
Go to the Google Drive for ALL course material.
Announcements
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Go to the Google Drive for ALL course material.
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