Algebraic Structures, Fall 2024

What:

The aim of this course is to introduce algebraic structures that pervade mathematics: groups and rings. We will learn what they are, will see many examples, learn how to reason about them. Topics to be covered include permutation groups, abelian groups, cyclic groups, homomorphisms, quotient groups, Sylow theorems, group classification, rings, ring homomorphisms, ideals, integral domains, quotient rings, unique factorization domains, principal ideal domains, and fields.

The prerequisites are being comfortable with reading and writing proofs, and a little of bit of linear algebra.

Textbook:

• Topics in Algebra 2nd ed. by Herstein. A copy is on reserve in the library.
• Read the book. It is a must for success in this course.
• Doing book exercises not assigned as homework is an excellent self-practice. Feel free to discuss these on Piazza.

Office hours:

The office hours will be at 1:00pm–1:55pm on Mondays and 2:00pm–2:50pm on Thursdays in Wean 6202. I am also available by appointment.

TA office hours:

We have two teaching assistants. Each of which holds the office hours, as follows:

Rui Zhou	Thursdays 4pm–5pm	Wean 7201
Zelong Li	Fridays 2pm–3pm	Wean 7102

Discussion forum:

We use Piazza discussion forum.

Course activities:

There will be weekly homeworks, two mid-terms and a final. The tentative mid-term dates are October 2nd and November 13th. The final exam has been scheduled by registrar for December 15th at 1pm. In case of a final exam conflict, the students are required to inform me and the other involved instructor within one week after the registrar publishes the final exam schedule.

Students are expected to fully participate in the class. To avoid distractions, use of cellular phones (texting, calling, etc) during class is not allowed. Discussions during the lectures are encouraged.

Homework will count for 10% of the grade. The mid-terms will count for 25% each, whereas the final will count for 40%.

Homeworks:

Practice is an integral to learning mathematics. You are encouraged to do as much homework as possible on your own; this way you will learn more. Though collaboration is allowed, you must write the solutions yourself. Turning in solutions that you do not understand will be treated as cheating. In particular, you are allowed to use (with a citation) any source, but only if you have read and understood it.

Homework must be neat. Each word must be readable. Anything that you do not want to be graded must be completely crossed out. If in doubt, either re-write solution from scratch or typeset it in LaTeX. Any solution that fails to be neat will receive 0.

If you want to dispute a homework grade, you should do it within five days from the time the grades were released.

The two lowest homework scores will not count towards the final grade.

The homework must be submitted via Gradescope.

Exams:

All exams are closed book.

If for unforeseeable reason you are unable to take one of the exams, contact me as soon as you are able.

Academic integrity:

Violations of academic integrity include, but are not limited to,

• Not writing solutions independently.
• Not disclosing help that you received and resources used.
• Turning in solutions that you do not understand.
• Receiving or providing assistance during an exam.

The default course-level action for violation of academic integrity is an R grade for the course, but more lenient action may be considered depending on the nature of the violation and conduct during the investigation.

If you feel desparate, and are tempted to commit a violation, note that there is probably a better way. Please reach out for support to me, and or to many of the university-wide resources.

Staying sane and healthy:

This is an advanced mathematics course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.

Lectures:

• August 26: Introduction. Groups. Examples. Commutativity. Abelian groups. Subgroups (taster). Homework #1
• August 28: Concatenation notation. Basic properties. Subgroups (definition). Examples. Cyclic subgroups. Congruence.
• August 30: Right cosets. Left cosets. Order of a group. Index. Lagrange's theorem. Groups of prime order are cyclic.
• September 2: Labor Day Homework #2
• September 4: Left index and right index. Order of an element. Euler's theorem. Euler's phi-function. Product of subgroups (part I).
• September 6: Product of subgroups (part II). Normal subgroups. Examples of normal and non-normal subgroups. Quotient groups.
• September 9: Examples of quotients. Homomorphisms. Examples. Natural projection. Isomorphisms. Kernel. Homework #3
• September 11: Trivial kernels. Kernels and quotients. Cauchy's theorem for abelian groups.
• September 13: More examples of quotients. Automorphisms. Conjugation. Inner automorphisms. Automorphism group. Inner automorphism group.
• September 16: Center. Automorphisms of cyclic groups. Cayley's theorem. Homework #4
• September 18: Cycles. Permutations are products of disjoint cycles. Transpositions. Sign of a permutation (part I).
• September 20: Sign of a permutation (part II). Alternating group. Conjugation in S_n. Normalizers.
• September 23: Size of the conjugacy class. The class equation. Cauchy's theorem. Groups of order p^m have non-trivial center. Groups of order p². Sylow I. Retelling of the third proof of Sylow's theorem. Homework #5
• September 25: p-Sylow subgroups. Statements of Sylow II and Sylow III. Normalizers of subgroups. Conjugacy relative to a subgroup. A much-needed lemma. Proof of Sylow II and III (very beginning).
• September 27: Proof of Sylow II and III. Using Sylow's theorem to find normal subgroups.
• September 30: More applications of Sylow. Direct products (both external and internal). Homework #6
• October 2: Midterm #1
• October 4: Q is not a product of cyclic groups. Classification of finite abelian groups.
• October 7: Uniqueness of factorization into cyclic groups. Homework #7
• October 9: Rings. Examples. Fields. Zero divisors. Integral domains. Finite integral domains are fields.
• October 11: Products of rings. Complex numbers. Homomorphisms of rings. Examples of homomorphisms. Subrings. Isomorphisms of rings. Kernels.
• October 21: Ideals. Intersection and product of ideals. Quotients of rings. Examples. Homework #8
• October 23: Sums of ideals. Chinese remainder theorem. Ideals in quotients of rings. Maximal ideals. Examples of maximal ideals.
• October 25: Chinese remainder theorem in polynomial rings. Fields of fractions. Euclidean domains. Examples of Euclidean domains. Division of polynomials.
• October 28: Prime elements. Units. Associates. Factorization in Euclidean domains. Homework #9
• October 30: Uniqueness of factorization in Euclidean domains. Gaussian integers.
• November 1: Primes that are sums of two squares. Irreducible polynomials.
• November 4: Complex numbers as a quotient ring. Some finite fields. Primitive polynomials. Content. Gauss's lemma. Factorization in Q[x]. Eisenstein's criterion. Homework #10
• November 6: Polynomials in several variables. Unique factorization domains. Polynomials rings over UFDs.
• November 8: Review of vector spaces over fields. Examples. Subfields. Field extensions. Degree. Finite extensions. Degree of iterated extension.
• November 11: Algebraic elements. Minimal polynomial. Adding a root. Homework #11
• November 13: Midterm #2
• November 15: Adding one root vs several. Algebraic extensions. Iterated algebraic extensions. Q-bar.
• November 18: Algebraically closed fields. Fundamental theorem of algebra (no proof). Root multiplicity. Finite multiplicative subgroups of a field. Splitting fields (definition and existence). Homework #12
• November 20: Splitting fields (uniqueness). Straightedge and compass constructions, part I.
• November 22: Straightedge and compass constructions, part II. Irrationality of e and related numbers.
• November 25: Distinct roots. Formal derivatives. Construction of finite fields. Homework #13
• December 2: Simple extensions. Finite extensions in characteristic 0. Field automorphisms. Fixed field. Examples.
• December 4: Linear independence of field automorphisms. Symmetric functions. Elementary symmetric functions.
• December 6: Normal extensions. Fixed subfields of normal extensions. A short overview.
• December 15: Final exam Two 85-minute long parts, separated by a 10 minute break. List of possible theorems-from-the-course-to-prove on the final exam