Math 404A: Introduction to Modern Algebra

Spring 2021

Lectures: MWF 10:30-11:20 on Zoom
Instructor: Jarod Alper (
Office hours: Wed 3-4 pm on zoom


Course content: The course will cover Galois theory, one of the crown jewels of modern mathematics. Galois’s mathematical approach in the early 19th century was strongly influenced by Abel, Bernoulli, Euler, Gauss, Legendre, Lagrange, Newton and many others stretching back to antiquity. Galois's ideas exerted an extraordinary influence on the course of modern mathematics, shifting emphasis away from explicit, messy algebraic calculations, and towards an understanding of the abstract structures that lay behind them. Using the techniques developed in this course, we will solve problems which had been open for thousands of years!

Schedule: A very brief summary of the content of each lecture will be posted here. The sections of the textbook quoted below are only a rough approximation of what we actually covered in class.

Date Topics covered Due Video, slides and notes
Week 1
Mon Mar 29 Introduction: course summary and policies
Wed Mar 31 Cubic equations Reflection 1 video, slides
notes on cubics (Smyth)
Fri Apr 2 Cubic equations continued HW 1
HW1 solns
video, slides
Quick history of polynomials (T. Bösel)
Biography of del Ferro (St. Andrews collection)
• Cubic dispute: L. Liberti, St. Andrews collection.
Week 2
Mon Apr 5 Review of group actions, structure of roots of polynomials in k[x], symmetric polynomials video, slides
Chapter 14--Group Actions (Judson)
notes on roots and symmetric polynomials (Smyth)
Wed Apr 7 Elementary symmetric functions, Fundamental Theorem on Symmetric Functions, Lagrange's solution to the quartic Reflection 2 video, slides
Lagrange's quartic solution (Smyth)
Fri Apr 9 Recap Lagrange's solution to quartic, review of groups, rings & fields HW 2
HW 2 solns
video, slides
Week 3
Mon Apr 12 Review of vector spaces and basis, field extensions (degrees and simple field extensions) Quiz 1 video, slides
• Hungerford 11.1-11.2
Wed Apr 14 Simple field extensions, algebraic elements, minimal polynomials Reflection 3 video, slides
• Hungerford 11.2-11.3
Fri Apr 16 Algebraic elements, algebraic field extensions, degrees of field extensions in towers HW 3
HW 3 solns
video, slides
• Hungerford 11.1-11.3
Week 4
Mon Apr 19 Finite, algebraic and transcendental field extensions; sums and products of algebraic elements are algebraic Quiz 2 video, slides
• Hungerford 11.3
Wed Apr 21 Ruler and compass constructions I Reflection 4 video, slides
• Hungerford 15
Fri Apr 23 Ruler and compass constructions II HW 4
HW 4 solns
video, slides
• Hungerford 15
Week 5
Mon Apr 26 Splitting fields Quiz 3 video, slides
• Hungerford 11.4
Wed Apr 28 Uniqueness of splitting fields and discussion Reflection 5 video, slides
• Hungerford 11.4
Fri Apr 30 Normal field extensions, splitting fields are normal, definition of separable field extensions HW 5
HW 5 solns
video, slides
• Hungerford 11.4-11.5
Week 6
Mon May 3 Separable field extensions Quiz 4 video, slides
• Hungerford 11.5
Wed May 5 Finite fields Reflection 6 video, slides
• Hungerford 11.6
Fri May 7 Revisiting normal and separable field extensions; further discussion of finite fields HW 6
HW 6 solns
video, slides
• Hungerford 11.4-11.6
Week 7
Mon May 10 First properties of the Galois group Quiz 5 Galois bio (MacTutor)
Mirzakhani bio (MacTutor)
The Beautiful Mathematical Explorations of Maryam Mirzakhani (Quanta magazine)
video, slides
• Hungerford 12.1
Wed May 12 Examples of Galois groups and discussion Reflection 7 video, slides
• Hungerford 12.1
Fri May 14 More on Galois groups, statement of the Fundamental Theorem of Galois Theory HW7
HW 7 solns
video, slides
• Hungerford 12.2
Week 8
Mon May 17 Fundamental Theorem of Galois Theory Quiz 6 video, slides
• Hungerford 12.2
Wed May 19 Discussion of Galois theory and homework Reflection 8 video, slides
• Hungerford 12.2
Fri May 21 Wrapping up the Fundamental Theorem of Galois Theory: For K -> E-> L, E is normal over K <=> Gal(L/E) is normal in Gal(L/K) HW8
HW8 solns
video, slides
• Hungerford 12.2
Week 9
Mon May 24 Galois's criterion Quiz 7 video, slides
• Hungerford 12.3
Wed May 26 Discussion Reflection 9 video, slides
Fri May 28 Galois's criterion HW9
HW9 solns
video, slides
• Hungerford 12.3
Week 10
Mon May 31 No class--Memorial Day
Wed Jun 2 Wrapping up Galois's criterion video, slides
• Hungerford 12.3
Fri Jun 4 Discussion HW10 not recorded

Homework assignments: There will be 10 weekly homework assignments each submitted on Canvas. Homeworks will be due by 11 pm each Friday. The homework assignments constitute a large component of the class. Homework provides you an opportunity to engage directly with the course material and reinforce ideas from the lectures and textbook.

A selection of the homework problems will be graded and returned to you. Your homework average contributes to 55% of your final grade.

You should expect to spend a considerable amount of time working on the homework. Typically you may want to spend at least six hours outside of class on each homework.

The lowest homework score will be dropped. Homework extensions will not be granted except for extraordinary circumstances.

Reflections: Self-reflections can serve as important educational devices. You are required to submit through Canvas a short self-reflection journal entry documenting your mathematical journey over the last week. Self-reflections are due 11 pm each Wednesday. These entries should be at least one or two paragraphs but you are welcome to write more. Reflections will be graded based on completion with a score of either 0 or 1. Your are allowed to miss one self-reflection.

The content of each self-reflection is up to you but here are some ideas for topics you may want to address:

Quizzes: There will be 5-8 quizzes during the course each administered during your own time on Canvas. Each quiz is designed to take 20 minutes but you will be given a total of 30 minutes to compensate for the additional time needed to download/upload the quiz. The lowest quiz score will be dropped.

The content of each quiz will be directly based on the previous one or two homework assignments.

Course policies: