MATH 4000/6000: Modern Algebra & Geometry I

MWF 11:30 AM-12:20 PM, Boyd Research and Education Center, Room 304
Spring 2024
406 Boyd Graduate Studies Building
Office hours:
Monday, 3-5 PM
Wednesday, 3-4 PM
Friday, 3-4 PM
This page is a copy of the syllabus. The web page will occasionally be updated with additional course materials.
Assignments and other course material
  • Final exam learning objectives/study guidePDF
  • Exam #3 learning objectives/study guidePDF
  • Homework #7due 4/17PDF
  • Homework #6due 4/8PDF
  • Exam #2 learning objectives/study guidePDF
  • Homework #5due 3/18 3/20PDF
  • Homework #4 (updated 2/27)3/1, by 5 PM PDF
  • Exam #1 review PDF
  • Homework #3due 2/9 2/12 PDF
  • Homework #2due 1/31PDF
  • Homework #1due 1/22PDF
  • $\mathbb{Z}$ axioms handoutPDF
  • Printable syllabusDOCX
Semester-to-date summary (reverse chronological order)
  • 4/29 For every prime $p$ and each positive integer $n$, there is a field of size $p^n$. Every finite field has size $p^n$ for some prime $p$ and some positive integer $n$. Bonus theorem (no proof): Any two finite fields of the same size are isomorphic.
  • 4/26 If $[K:F]=n$, and $\alpha \in K$ is a root of the irreducible polynomial $p(x) \in F[x]$, then $\deg{p(x)}$ divides $n$. If $[K:F]$ is prime, and $\alpha \in K\setminus{F}$, then $K = F[\alpha]$. Examples of degree problems.
  • 4/24 The degree is multiplicative "in towers." If $K$ is an extension of $F$ of finite degree, every element of $K$ is algebraic over $F$.
  • 4/22 Introduction to degrees of field extensions. Examples.
  • 4/19 Exam #3.
  • 4/17 More on splitting fields.
  • 4/15 If $F$ is a field and $f(x) \in F[x]$ is nonconstant, there is an extension $K$ of $F$ where $f$ has a root. If $F$ is a field and $f(x) \in F[x]$ is nonconstant, there is an extension $K$ of $F$ where $f$ splits. Example of constructing a splitting field for $x^3-2$ over $\mathbb{Q}$.
  • 4/12 Proof of the Fundamental Homomorphism Theorem. Definition of splitting over a field $F$. Statement of the Fundamental Theorem of Algebra. Example of constructing a field where a given polynomial has a root.
  • 4/10 $\mathbb{Z}_{mn}\not\cong \mathbb{Z}_m\times \mathbb{Z}_n$ when $\gcd(m,n)=1$. Statement of the Fundamental Homomorphism Theorem. $\mathbb{Z}_{10}/\{0,5\} \cong \mathbb{Z}_5$. If $F$ is a subfield in $K$, and $\alpha \in K$ is a root of the irreducible polynomial $p(x) \in F[x]$, then $F[x]/\langle p(x)\rangle \cong F[\alpha]$.
  • 4/8 Direct product of rings. $\mathbb{Z}_{mn} \cong \mathbb{Z}_m \times \mathbb{Z}_n$ when $m$ and $n$ are relatively prime.
  • 4/5 Introduction to isomorphisms. Conjugation is a nontrivial automorphism of $\mathbb{C}$. There is no nontrivial autmorphism of $\mathbb{Z}_p$. There is a nontrivial automorphism of $\mathbb{Z}_2[x]/\langle x^2+x+1\rangle$. The Freshman's Dream!
  • 4/3 Computing in $F[x]/\langle f(x)\rangle$. If $f(x)$ is reducible, then $F[x]/\langle f(x)\rangle$ is not a domain. If $f(x)$ is irreducible, then $F[x]/\langle f(x)\rangle$ is a field.
  • 4/1 Construction of the quotient ring $R/I$. Example of $F[x]/\langle f(x)\rangle$. If $f$ has degree $n\ge 1$, then every element of $F[x]/\langle f(x)\rangle$ has the form $\overline{a_0+a_1 x + \dots + a_{n-1} x^{n-1}}$ for some $a_i \in F$.
  • 3/29 Every ideal of $\mathbb{Z}$ is principal. Every ideal of $F[x]$ ($F$ a field) is principal. Not every ideal of $\mathbb{Z}[x]$ is principal. Every ideal of $\mathbb{Z}$ is the kernel of a homomorphism from $\mathbb{Z}$ to somewhere.
  • 3/27 Exam #2.
  • 3/25 Definition of a homomorphism. Homomorphisms send $0$ to $0$ and preserve additive inverses. The kernel is trivial if and only if the homomorphism is injective. Definition of an ideal of a (commutative) ring.
  • 3/22 Testing irreducibility by reducing modulo $p$. Eisenstein's irreducibility criterion.
  • 3/20 Adjoining any finite number of algebraic elements to a field $F$ gives a field. Start of discussion of irreducibility over $\mathbb{Q}$. Rational root test and proof. Statement of Gauss's lemma.
  • 3/18 The rings $F[\alpha,\beta]$ and more generally $F[\alpha_1,\dots\alpha_n]$. Case study: $\mathbb{Q}[\sqrt{2},\sqrt{3}]$. Proof $\mathbb{Q}[\sqrt{2},\sqrt{3}]$ is a field.
  • 3/15 If $F\subseteq K$ and $\alpha \in K$ is algebraic over $F$, then there is an irreducible polynomial $p(x) \in F[x]$ with $p(\alpha)=0$. If $F\subseteq K$ and $\alpha \in K$ is algebraic over $F$, then $F[\alpha]$ is a field. Description of all elements of $F[\alpha]$.
  • 3/13 Examples of subfields. Definition of $F[\alpha]$, where $F \subseteq K$ and $\alpha \in K$. $F[\alpha]$ is a subring of $K$. Definition of being algebraic over $F$. Direct proof that $\mathbb{Q}[\sqrt{2}]$ is a field. Statement that $F[\alpha]$ is a field if and only if $\alpha$ is algebraic over $F$.
    • 3/11 Relatively prime polynomials. A worked example. Fundamental gcd lemma in $F[x]$. Euclid's lemma in $F[x]$. Definition of subfield/extension field.
    • 3/1 Statement of unique factorization of polynomials. Proof of existence. Statement of Euclid's lemma. Definition of $\gcd$ in $F[x]$. Existence via Euclidean algorithm. Expression of the gcd as a linear combination of the starting polynomials.
    • 2/28 Long division in $F[x]$ (with $F$ a field). Root factor theorem. Remainder theorem. Definition of irreducible polynomials in $F[x]$.
    • 2/26 Introduction to polynomials over commutative rings $R$ (not the zero ring). $R[x]$ is a ring. $R[x]$ is a domain when $R$ is a domain; in this case, $\deg{a(x)b(x)}=\deg{a(x)}+\deg{b(x)}$. Statement of division algorithm in $F[x]$ ($F$ a field).
    • 2/21 How to construct $\mathbb{R}$, continued (non-examinable).
    • 2/19 There is a rational number strictly between any two rational numbers. The rational numbers are "Archimedean". How to construct $\mathbb{R}$ (non-examinable).
    • 2/16 Exam #1.
    • 2/14 $\mathbb{Q}$ contains a copy of $\mathbb{Z}$. $\mathbb{Q}$ is an ordered field. There is no smallest positive element in $\mathbb{Q}$.
    • 2/12 Completion of the construction of $\mathbb{Q}$. $\mathbb{Q}$ is a commutative ring, in fact a field. Start of discussion of how $\mathbb{Q}$ contains a copy of $\mathbb{Z}$.
    • 2/9 Zero divisors. Examples of proofs from the ring axioms. Introduction to the construction of $\mathbb{Q}$.
    • 2/7 Definition of the term unit. $\mathbb{Z}_p$ is a field whenever $p$ is prime. Description of all units in $\mathbb{Z}_m$. Analyzing integer solutions to equations by working in $\mathbb{Z}_m$.
    • 2/5 Definition of the term integral domain (or domain). $\mathbb{Z}_m$ is the zero ring if and only if $m=1$. $\mathbb{Z}_m$ is an integral domain if and only if $m$ is prime. Definition of the term field. Every field is an integral domain.
    • 2/2 Definition of $\mathbb{Z}_m$. Definitions of the terms ring and commutative ring. $\mathbb{Z}_m$ is a commutative ring. There is a unique `zero ring'.
    • 1/31 Systems of congruences and the Chinese remainder theorem.
    • 1/29 The existence of "$1/a$ mod $m$" whenever $\gcd(a,m)=1$. Solving linear congruences.
    • 1/26 Each integer is congruent, mod $m$, to a unique integer from $0,1,\dots,m-1$. Examples of computing with congruences. Middle binomial coefficients in the $p$th row are multiples of $p$ (when $p$ is prime).Fermat's little theorem.
    • 1/24 Euclid's lemma. Proof of uniqueness of prime factorization. Definition of congruences. Congruence mod $m$ is an equivalence relation on $\mathbb{Z}$. Congruence mod $m$ plays nice with addition and multiplication.
    • 1/22 Correctness of Euclid's algorithm. Every common divisor divides the gcd. The gcd as a linear combination of the starting numbers. Fundamental gcd lemma. Every integer $>1$ is a product of prime numbers.
    • 1/19 Proof of the division algorithm. Definition of $d\mid n$, along with examples and basic properties. Definition of $\gcd$ and Euclid's magic gcd algorithm. Euclid's algorithm stops.
    • 1/17 Exponentiation. Pascal's triangle and the binomial theorem. Statement of Division algorithm.
    • 1/12 More deductions from the fundamental properties: $a(-1)=-a$ and $(-1)a=-a$, rules for inequalities, $1$ is the least positive integer. Principle of Mathematical Induction as a consequence of WOP.
    • 1/10 Fundamental Properties of $\mathbb{Z}$, continued. Proofs that $-(-a)=a$ and $a \cdot 0 = 0$.
    • 1/8Review of syllabus. Fundamental properties of $\mathbb{Z}$.
Course synopsis

At this point in your mathematical career, you have accumulated a wealth of experience computing with integers, real numbers, and complex numbers. These items are so familiar to you that they may even have the appearance of being God-given --- as if understanding those objects is What Mathematics is All About.

Abstract algebra challenges this notion. Integers, real numbers, and complex numbers are indeed fantastic, but they are fantastic not because they are handed down from on high, but because they have a rich and (presumably) consistent theory with useful consequences. And given how useful these objects are, we are compelled to isolate (and consider in the abstract) their most important properties. Once we do so, we find that there are many other objects with the same sorts of interesting properties. For example, we will see that the integers are an example of what is called a ring, while the real numbers and complex numbers are fields. These more abstract-seeming objects are not just interesting in an intellectual let's-talk-about-this-over-coffee kind of way, but understanding them deeply often leads to a new understanding of the objects of original interest.

Let me give one concrete example: Some primes, like 5, can be written as a sum of two squares: $5 = 1^2 + 2^2$. And other primes, like 3, cannot. The question of when this is possible is a question about the integers. But our answer to this question --- and we will answer it by the end of the semester --- will require us to visit a totally different mathematically system, the ring of Gaussian integers.

Textbook (not required)

Topics to be covered include integers and unique factorization, modular arithmetic, analogues of these for polynomials over fields, homomorphisms and quotient rings, and field extensions.

We will aim to cover Chapters 1--4 of Shifrin's text and to start Chapter 5 (integers, modular arithmetic, analogues for polynomials over fields, homomorphisms and quotient rings, and field extensions). This corresponds roughly to Chapters 1--6 and some of Chapter 11 in Hungerford's book. For further adventures in Algebra-land, I encourage you to take MATH 4010.

Homework/ Exams/ Grading

There will be three fifty-minute in-class exams, as well as a final exam.

  • Midterm #1: Friday, Feb. 16
  • Midterm #2: Wednesday, March 27
  • Midterm #3: Friday, April 19
  • Final exam: Friday, May 3 12:00 - 3:00 PM (usual classroom)

No make-up exams will be given. The final exam is cumulative. Your grade is made up of the following weighted components:

  • Each midterm: 15% (total of 45%)
  • Homework: 25%
  • Final exam: 30%

This class falls into the interactive lecture genre (not entirely unrelated to the practice of call and response in a liturgical context). What this means is that I intend to punctuate the lectures frequently with questions for you. For the show to go on, class participation is absolutely essential. Since you cannot participate in class if you are not present in class, your attendance is required. In particular, more than four unexcused absences may result in you being automagically withdrawn from the class. Of course, missing class is sometimes a necessity; keep me posted whenever you have a conflict and we should not have any issues.

Homework will be collected roughly once each week. As a general rule, late homework assignments are not accepted. Your lowest HW score will be dropped at the end of the semester.

All exams are closed book and closed notes.

Students enrolled in MATH 6000 will take the same exams as the students in MATH 4000 but will be assigned additional homework problems.

You are not only allowed, but encouraged to collaborate with your classmates on the homework assignments. The joy of mathematical discovery was meant to be shared! Having said that, collaboration does not mean copying. You may not copy solutions from a textbook, classmate, website, etcetera, and you must be the one to handwrite (or type) your solutions. By entering UGA, you have already agreed to abide by the UGA honor code described in detail at A good rule of thumb is that you should be able to explain any work you turn in to a hypothetical interrogator.

Office hours

Mathematics is not a spectator sport! One learns mathematics by doing it. What I'll be sharing with you in class can be thought of as a hindsight-informed summary of productive struggle by many mathematics over several centuries. Just as it's impossible to absorb the material in a textbook by sleeping with the book under your pillow, it's impossible for you to come to grips with the results of their hard labor by sitting in lectures. Thinking hard about problems --- and being stumped a good deal of the time! --- is part and parcel of coming into your own as a mathematical person.

Office hours exist to make this struggle a little more pleasant. They are intended to function as a safe space for you to share ideas and to work through the homework, with the support of both myself and your peers. I strongly encourage you attend.

Both at office hours and in class, you should expect that your input will be treated respectfully, by myself and by your classmates. Turning it around, you are expected to show respect and understanding for your classmates’ ideas. Kindness is important, everywhere and always!

Special accommodations

Students with disabilities who may require special accommodations should talk to me as soon as possible. Appropriate documentation concerning disabilities may be required. For further information, please visit the Disability Resource Center page.

Mental health and wellness

If you or someone you know needs assistance, you are encouraged to contact Student Care and Outreach in the Division of Student Affairs at 706-542-8479 or visit They will help you navigate any difficult circumstances you may be facing by connecting you with the appropriate resources or services. UGA has several resources for a student seeking mental health services ( or crisis support (

FERPA Notice

The Federal Family Educational Rights and Privacy Act (FERPA) grants students certain information privacy rights. See the registrar’s explanation at FERPA allows disclosure of directory information (name, address, telephone, email, major, activities, degrees, awards, prior schools), unless requested in a written letter to the registrar.


The course syllabus is a general plan for the course; deviations announced to the class by the instructor may be necessary.