Math 5111 (Algebra I), Fall 2020
| Course Information | |||||
|---|---|---|---|---|---|
| Instructor | Class Times | Office Hours | |||
| Evan Dummit edummit at northeastern dot edu |
MR 6:00pm-7:30pm Online, via Zoom |
MWR 3:00pm-4:15pm or by appointment Online, via Zoom |
|||
| All lectures will be recorded and made available for on-demand viewing at any time. You are encouraged to attend and participate in the lectures in real time, but this is not required. | |||||
| Math 5111 uses a Piazza page for course discussion. Links to all of the live lectures, office hours, and lecture recordings are hosted there. | |||||
| For detailed information about the course, please consult the 5111 Course Syllabus. (Note: any information given in class or on this webpage supersedes the written syllabus.) | |||||
| Homework Assignments & Exams |
|---|
|
Homework #1, due Fri Sep 18th. (solutions)
Homework #2, due Fri Sep 25th. (solutions) Homework #3, due Fri Oct 2nd. (solutions) Homework #4, due Fri Oct 9th. (solutions) Homework #5, due Fri Oct 16th. (solutions) Homework #6, due Fri Oct 23rd. (solutions) Homework #7, due Fri Oct 30th. (solutions) Homework #8, due Fri Nov 6th. (solutions) Midterm Exam, due Fri Nov 13th. (solutions) Homework #9, due Fri Nov 20th. (solutions) Homework #10, due Fri Nov 27th. (solutions) Homework #11, due Fri Dec 4th. (solutions) Homework #12, due Fri Dec 11th. (solutions). Here is an abbreviated version of the transitive subgroups tables from the notes, which also has information about solvability and which groups are subgroups of An. Here is the Frobenius-counting Mathematica code. Final Exam, due Fri Dec 18th. Like the midterm, the final is take-home and open book, open notes, etc. |
| Handouts / Lecture Notes | |||||
|---|---|---|---|---|---|
| The instructor will write lecture notes for the course in place of an official textbook as the semester progresses. The course will follow the presentation in Dummit and Foote's "Abstract Algebra" (3th edition), and you are highly encouraged to purchase this book since it is an excellent general reference for algebra. Here are approximate correspondences between the notes and textbook:
Notes ch1: Dummit/Foote 0.1-0.3 + 9.1-9.5 + 7.1-7.3 Notes ch2: Dummit/Foote 11.1 + 13.1-13.5 Notes ch3: Dummit/Foote 1.1-1.7 + 2.1-2.5 + 3.1-3.3 + 3.5 + 4.1-4.6 + 5.1-5.5 Notes ch4: Dummit/Foote 14.1-14.2 + 13.6 + 14.3-14.9 |
|||||
| Handout | Topics | ||||
| Chapter 1: Polynomials and Rings (38pp, v2.00, posted 9/5) | 1.1 ~ Integers and Modular Arithmetic (Divisibility, GCDs, Euclidean Algorithm, Prime Factorization, Congruences, Z/mZ) 1.2 ~ Polynomials (Division Algorithm, Euclidean Algorithm, Irreducibility, Unique Factorization, Modular Arithmetic) 1.3 ~ Survey of Rings (Definition, Examples, Basic Properties, Ideals, Quotient Rings, Isomorphisms, Homomorphisms, Ideals and Homomorphisms) |
||||
| Chapter 2: Fields and Field Extensions (40pp, v2.20, updated 10/4) (updates in 2.20: moved transcendence to 2.4.3, added 2.4.2, added paragraph at end of 2.4.3) (updates in 2.10: added 2.4, moved separability into 2.4.1, minor typo fixes) |
2.1 ~ Fields and Vector Spaces 2.2 ~ Subfields and Field Extensions (Examples, Properties, Simple Extensions, Algebraic Extensions, Small-Degree Examples, Geometric Constructions) 2.3 ~ Splitting Fields (Properties, Examples, Algebraic Closures) 2.4 ~ Separability and Transcendence |
||||
| Chapter 3: Groups (49pp, v2.00, updated 11/2) (updates in 1.80: fixed typos, added 3.4.1-3.4.2) (updates in 2.00: added 3.4.3-3.4.4) |
3.1 ~ Examples and Basic Properties of Groups (Dihedral Groups, Symmetric Groups, Subgroups, Orders of Elements, Generation and Presentations, Cyclic Groups, Isomorphisms and Homomorphisms) 3.2 ~ Cosets and Quotient Groups (Cosets of Subgroups and Lagrange's Theorem, Normal Subgroups and Quotient Groups, Quotients and Homomorphisms) 3.3 ~ Group Actions (Definition and Basic Properties, Polynomial Invariants and An, Conjugation Actions) 3.4 ~ The Structure of Finite Groups (Abelian Groups, Sylow's Theorems, Products of Subgroups, Semidirect Products) |
||||
| Chapter 4: Galois Theory (48pp, v1.50, posted 11/9) |
4.1 ~ Field Automorphisms and Galois Groups (Field Automorphisms, Computing Automorphisms, Automorphisms of Splitting Fields and Galois Groups, Fixed Fields) 4.2 ~ The Fundamental Theorem of Galois Theory (Characterizations of Galois Extensions, Proof of the Fundamental Theorem, Examples) 4.3 ~ Applications of Galois Theory (Finite Fields, Simple Extensions, Composite Extensions, Cyclotomic Extensions) 4.4 ~ Galois Groups and Polynomials (Symmetric Functions, Discriminants, Cubic Equations, Quartic Equations, Computing Galois Groups over Q, Solvability in Radicals) |
||||
| Lecture Slides | |||
|---|---|---|---|
| These are the slides used during the lectures. They will usually be posted ahead of the lecture time. | |||
| Date | Material | ||
| Thu, Sep 10th | Lecture 1: Integers and Polynomials (Notes 1.1-1.2.4) | ||
| Mon, Sep 14th Thu, Sep 17th |
Lecture 2: Polynomials 2, Rings 1 (Notes 1.2.4-1.3.2)
Lecture 3: Rings 2 (Notes 1.3.3-1.3.8) | ||
| Mon, Sep 21th Thu, Sep 24th |
Lecture 4: Fields and Vector Spaces (Notes 2.1)
Lecture 5: Subfields and Simple Extensions (Notes 2.2.1-2.2.3) | ||
| Mon, Sep 28th Thu, Oct 1st |
Lecture 6: Algebraic Extensions (Notes 2.2.4-2.2.5)
Lecture 7: Classical Constructions + Splitting Fields 1 (Notes 2.2.6-2.3.2) | ||
| Mon, Oct 5th Thu, Oct 8th |
Lecture 8: Splitting Fields 2 + Algebraic Closures + Separability 1 (Notes 2.3.2-2.4.1)
Lecture 9: Separability 2 (Notes 2.4.1-2.4.2) | ||
| Mon, Oct 12th Thu, Oct 15th |
No class; university holiday.
Lecture 10: Separability 3, Transcendence (Notes 2.4.2-2.4.3) | ||
| Mon, Oct 19th Thu, Oct 24th |
Lecture 11: Basic Groups, part 1 (Notes 3.1.1-3.1.4)
Lecture 12: Basic Groups, part 2 (Notes 3.1.5-3.1.7) | ||
| Mon, Oct 26th Thu, Oct 29th |
Lecture 13: Cosets and Quotient Groups (Notes 3.2.1-3.2.2)
Lecture 14: Isomorphism Theorems + Group Actions (Notes 3.2.3-3.3.1) | ||
| Mon, Nov 2nd Thu, Nov 5th |
Lecture 15: An, Conjugation, and Abelian Groups (Notes 3.3.2-3.4.1)
Lecture 16: Abelian Groups + Sylow's Theorems (Notes 3.4.1-3.4.2) | ||
| Mon, Nov 9th Thu, Nov 12th |
Lecture 17: Products of Subgroups, Semidirect Products (Notes 3.4.3-3.4.4)
Lecture 18: More Semidirect Products, Field Automorphisms (Notes 3.4.4 + 4.1.1-4.1.2) | ||
| Mon, Nov 16th Thu, Nov 19th |
Lecture 19: Galois Groups and Fixed Fields (Notes 4.1.3-4.2.1)
Lecture 20: The Fundamental Theorem of Galois Theory (Notes 4.2) | ||
| Mon, Nov 23rd Thu, Nov 26th |
Lecture 21: Finite Fields, Primitive Elements, Composite Extensions (Notes 4.3.1-4.3.4)
No class, university holiday: happy Thanksgiving! | ||
| Mon, Nov 30th Thu, Dec 3rd |
Lecture 22: Galois Groups of Polynomials (Notes 4.3.4-4.4.3)
Lecture 23: Computing Galois Groups of Polynomials (Notes 4.4.4-4.4.5) | ||
| Mon, Dec 7th Thu, Dec 10th |
Lecture 24: Solvability in Radicals (Notes 4.4.6)
No more lectures: the course is over :-( | ||
| Tips For Success In This Course | |||
|---|---|---|---|
| Attend Lecture | Missing lecture is a bad idea! If for any reason you cannot make it to a class, you should watch the recording of the lecture. You are responsible for all material covered in lecture. | ||
| Read the Lecture Notes (or Textbook) | The lecture notes and the textbook are comprehensive sources of material for the course. The notes are intended as review material, although many students like to read them as preparation before attending the lecture on the corresponding topics. Please note that the electronic notes are not identical to the material covered in class: this is by design, so as to provide you a slightly different perspective on the material. | ||
| Attend Office Hours | Office hours are specifically reserved for you to receive individual, one-on-one help from the instructor. Office hours will be the most effective when you have already put in effort to learn the material on your own (including trying to solve the homework problems), and when you come in with a list of specific questions or topics you are struggling with. | ||
| Work With Other Students | One of the best ways to learn mathematics is by collaboration. You are strongly encouraged to collaborate with other students in this class on the homework assignments. This is not a large class, so it is a very good idea to get to know your fellow students and find other people who are interested in working on the assignments together. Piazza is a good place to start finding people for such collaborations. | ||
| Course Schedule | |||
|---|---|---|---|
| The schedule is subject to change! All sections refer to the course lecture notes. | |||
| Week | Schedule | ||
| Week of Sep 7th (class starts 9/10) |
§1.1: The Integers. §1.2: Polynomials. No homework this week. |
||
| Week of Sep 14th | §1.2: Polynomials. §1.3: Survey of Rings Homework 1 due Friday, Sep 18th. |
||
| Week of Sep 21st | §2.1: Fields and Vector Spaces §2.2.1: Subfields §2.2.2: Simple Extensions, Minimal Polynomials Homework 2 due Friday, Sep 25th. |
||
| Week of Sep 28th | §2.2.3: Algebraic Extensions §2.2.4: Small-Degree Extensions §2.2.5: Classical Geometric Constructions §2.3.1: Splitting Fields §2.3.2: Examples of Splitting Fields Homework 3 due Friday, Oct 2nd. |
||
| Week of Oct 5th | §2.3.2: Examples of Splitting Fields §2.3.3: Algebraic Closures §2.4.1: Separability of Polynomials §2.4.2: Separability of Extensions Homework 4 due Friday, Oct 9th. |
||
| Week of Oct 12th (no class 10/12) |
§2.4.3: Transcendence Homework 5 due Friday, Oct 16th. |
||
| Week of Oct 19th | §3.1: Examples and Basic Properties of Groups Homework 6 due Friday, Oct 23rd. |
||
| Week of Oct 26th | §3.2: Cosets and Quotient Groups §3.3: Group Actions Homework 7 due Friday, Oct 30th. |
||
| Week of Nov 2nd | §3.3: Group Actions §3.4.1: Abelian Groups §3.4.2: Sylow's Theorems Homework 8 due Friday, Nov 6th |
||
| Week of Nov 9th | §3.4.3: Products of Subgroups §3.4.4: Semidirect Products §4.1.1: Field Automorphisms §4.1.2: Computing Automorphisms Midterm exam due Friday, Nov 13th |
||
| Week of Nov 16th | §4.1.3: Automorphisms of Splitting Fields, Galois Groups §4.1.4: Fixed Fields §4.2: The Fundamental Theorem of Galois Theory Homework 9 due Friday, Nov 20th |
||
| Week of Nov 23rd (no class 11/26) |
§4.3.1: Finite Fields §4.3.2: Simple Extensions and the Primitive Element Theorem Homework 10 due Friday, Nov 27th |
||
| Week of Nov 30th | §4.3.3: Composite Extensions §4.3.4: Cyclotomic and Abelian Extensions §4.4.1: Symmetric Functions §4.4.2: Discriminants of Polynomials §4.4.3: Cubic Polynomials §4.4.3: Quartic Polynomials Homework 11 due Friday, Dec 4th |
||
| Week of Dec 6th (class ends 12/6) |
§4.4.4: Computing Galois Groups over Q §4.4.5: Solvability in Radicals Homework 12 due Friday, Dec 11th |
||
| Week of Dec 13th | Final exam due Friday, Dec 18th | ||