Course description: Welcome to MATH 5111, this class covers the fundamentals of group theory and its applications in Galois theory. Topics include:
Groups and their structure: Basic examples, homomorphisms and isomorphisms, types of subgroups, classification theory
Rings and fields: Types of rings, polynomial rings and their properties/algorithms
Galois theory: Field extensions, the Galois group and its applications
References:
Abstract Algebra, Third Edition, by David S. Dummit and Richard M. Foote. (required)
Algebra, Third Edition, by Serge Lang
Fields and Galois Theory, by J. S. Milne https://www.jmilne.org/math/CourseNotes/FT.pdf
Galois Theory, by Emil Artin
Syllabus: pdf
Midterm: pdf | Wed Oct 26th. In-class written exam, closed notes/book. Lectures 1-12, Homework 1-4.
Final Exam: pdf | Due by Friday December 16th 2022, 5:00pm.
Homework:
Homework 1 : pdf | Due : 09/21/2022
Homework 2: pdf | Due : 09/28/2022
Homework 3: pdf | Due : 10/05/2022
Homework 4: pdf | Due : 10/19/2022 10/12/2022
Homework 5: pdf | Due : 11/09/2022
Homework 6: pdf | Due : 11/16/2022
Homework 7: pdf | Due : 11/30/2022
Homework 8: pdf | Due : 12/07/2022
Lectures:
Lecture 1 [09/07/2022]: What’s the course about (Groups, rings, fields and their extensions, intro to Galois theory and its applications), what is a Group.
Lecture 2 [09/12/2022] pdf: Examples of groups, in particular the Symmetric and Dihedral group. Dummit & Foote Ch. 1.1, 1.2 and 1.3
Lecture 3 [09/14/2022] pdf: Subgroups and homomorphisms. Examples and detection criterion — center, alternating group, group generated by a subset (e.g., cyclic). Dummit & Foote Ch. 1.6, 2.1, 2.2, 2.3 and 2.4
Lecture 4 [09/19/2022] pdf: More on homomorphisms (epi, mono, iso), Cayley’s theorem, group actions (stabilizer and centralier). Dummit & Foote Ch. 1.6, 1.7, 2.2,
Lecture 5 [09/21/2022] pdf: Cosets and Lagrange’s thm, Normal subgroups, Isomorphism thm I. Dummit & Foote Ch. 3.1, 3.2, 3.3, 4.1
Lecture 6 [09/26/2022] pdf: More on normality (the normalizer and equivalent conditions). Dummit & Foote Ch. 3.1
Lecture 7 [09/28/2022] pdf: The commutator, operations between subgroups, ISO thm II. Dummit & Foote Ch. 3.2, 3.3, 3.4
Lecture 8 [10/03/2022] pdf: ISO thm III, Kernel (faithfulness) of an action, the orbit/stabilizer formula, subgroups with minimal prime index are normal. Dummit & Foote Ch. 3.3, 4.1, 4.2
Lecture 9 [10/05/2022] pdf: Cauchy’s thm (proof = Abelian version), the class equation, p-groups have nontrivial center, conjugation in S_n, A_5 is simple. Dummit & Foote Ch. 3.4, 4.3
Lecture 10 [10/12/2022] pdf: The Sylow theorems, Dummit & Foote Ch. 4.5
Lecture 11 [10/17/2022] pdf: Review, classifying groups of small order
Lecture 12 [10/19/2022] pdf: Classification of Abelian p-groups and finite Abelian groups. Lang Ch. I.7, I.8
Lecture 13 [10/24/2022] pdf: Free Abelian groups and torsion. Lang Ch. I.7, I.8
Lecture 14 [10/31/2022] pdf: Classification of finitely generated Abelian groups + Rings and ring homomorphisms. Lang Ch. I.7, I.8 + Dummit & Foote Ch 7.1, 7.2, 7.3
Lecture 15 [11/02/2022] pdf: Integral domains, ideals (principal and maximal), quotients, ISO theorems (I, II, III) for Rings. Dummit & Foote Ch 7.3, 7.4
Lecture 16 [11/07/2022] pdf: Prime ideals, Euclidean Domains and Principal Ideal Domains. Dummit & Foote Ch. 7.4, 8.1, 8.2
Lecture 17 [11/09/2022] pdf: Prime elements, Unique Factorization Domains (UFD), adding a root to a field, Eisenstein Criterion. Dummit & Foote Ch. 8.3, 9.2, 9.3, 9.4, 9.5
Lecture 18 [11/14/2022] pdf: Roots and irreducibility (rational roots test, Gauss’ lemma, Eisenstein’s Theorem), Field extensions and adding roots. Dummit & Foote Ch. 9.4, 13.1.
Lecture 19 [11/16/2022] pdf: Degree of a field extension, algebraic vs transcendental elements and minimal polynomial, Dummit & Foote Ch. 13.1, 13.2.
Lecture 20 [11/21/2022] pdf: Splitting fields, Dummit & Foote Ch. 13.4,
Lecture 21 [11/28/2022] pdf: Algebraic closure, separable extensions, Normal extensions, finite fields 13.4, 13.5
Lecture 22 [11/30/2022] pdf: Galois extensions and the Galois group Dummit & Foote Ch. 14.1
Lecture 23 [12/05/2022] pdf: The field fixed by a subgroup, characterizations of being Galois (e.g., splitting field of a separable polynomial). Dummit & Foote Ch 14.2
Lecture 24 [12/07/2022] pdf: The Fundamental Theorem of Galois Theory, and applications: The fundamental theorem of algebra, solvability by radicals Dummit & Foote Ch. 14.2, 14.6, 14.7