What does that mean in Algebra 2

Lecture Algebra II (summer semester 2020)

Event:
Algebra 2 is offered as a 9 LP course as well as a 6 LP course for student teachers. The 6 LP event consists of the first two thirds of the 9 LP event.
Lectures: Prof Steffen Koenig
Exercises: Dr Apolonia Gottwald

Dates: will be announced with the content and exercise sheets,

Requirements: linear algebra 1 and 2, algebra (or algebra and number theory for teachers), enjoyment of mathematics.

Current format: Lecture theses are now available in PDF format under content. It is also indicated how many lectures the provided chapters roughly correspond to, so that the time required for processing can be estimated. In the elaboration, small tasks or questions are entered in red, the processing of which should help you work through. Additional tasks are also provided under Exercises. This event takes place online. Details on the online components can be found in the Iliad courses.

Active reading of mathematical texts:
At the World Speed ​​Reading Championship, participants manage around 1000 to 2000 words per minute and understand half of the text. A multiple world champion managed up to 4200 words per minute and understood 67% of the text. This is not about math books.
Reading a math book or a research article takes at least an hour per page, and reading a lecture script as provided below takes at least half an hour, usually longer. Reading means - like listening in the lecture - taking notes and working out details in parallel or in a second round and looking up the results used. And note what you did not understand and what you have to work on again.
After working through a proof and clarifying the details, one should understand the structure of the proof, understand where the assumptions were used and whether / why you really need them.
Often sections or chapters have a goal or a main result and it helps to get an idea of ​​it before going through it in detail and then to think from time to time whether you have come closer to the goal.
When you have worked through a chapter or section, try to summarize it: what was the problem, how was it solved? Which new terms are important? What examples illustrate the terms and results? Does this text continue something that has already been discussed or does it establish connections? A short summary in your own words increases the learning effect. If you like to use highlighters, you can use them here.
When working on the exercises you can see what you have to work through again. But you should always set yourself small tasks: What is an example of the definition you have just read and in which example is this property not fulfilled? Is the evidence you just read constructive, and if so, how does it work in an example?

Next to the script is a date by which you should have worked through the relevant material. Correspondingly, the submission date is given for exercise sheets,

Content:
Chapter 0. Review: Field Extensions and Polynomial Equations.
Writing down (until April 22) for chapter 0, for independent editing, with questions of understanding in the writing.
The references refer to the previous lecture, see Algebra (winter semester 2019/20), for which a short script is also available.

Chapter 1. Galois theory.
Galois theory is a key issue. This chapter is particularly important, but not particularly easy. Understanding Galois theory and then applying it later is an experience.

Writing down (first part) (up to 29.4.) For chapter 1, for independent editing, with questions of understanding in the writing.
Writing down (second part) (up to 4.5.) For chapter 1, for independent processing, with questions of understanding in the writing.

Here is a brief biography of Galois. And for comparison with the original Galois lecture.
For an introduction to Galois theory of differential equations, see an article Introduction to the Galois Theory of Linear Differential Equations by Michael F. Springer, and the book Galois Theory of Linear Differential Equations by Marius van der Put and Michael F. Singer.

Chapter 2. Some Applications of Galois Theory.
Writing down (until May 15) for chapter 2, for independent editing, with questions of understanding in the writing.

Chapter 3. Polynomial Equations.
Writing down (until May 29) for chapter 3, for independent editing, with questions of understanding in the writing.

Whitsun holidays: June 1-6.

Chapter 4. Review: vector spaces, Abelian groups, main ideal rings.
Writing down (until 8/8) for chapter 4, for independent editing, with questions of understanding in the writing. (Repetition of linear algebra and algebra.)

Chapter 5. Modules on main ideal rings.
Writing down (first part) (until 17.6.) For chapter 5, for independent editing, with questions of understanding in the writing.
Writing down (second part) (up to 1.7.) For chapter 5, for independent editing, with questions of understanding in the writing.

Chapter 6. Applications: Abelian Groups and Matrices.
Writing down (up to 2.7.) For chapter 6, for independent editing, with questions of understanding in the writing.

Chapter 7. Group Algebras and Characters.
Writing down (first part) (up to 8 July) for chapter 7, for independent editing, with questions of understanding in the writing.
Writing down (second part) (until July 11th) for chapter 7, for independent processing, with questions of understanding in the writing.

Chapter 8. Character panels.
Writing down (until July 14th) for chapter 8, for independent editing, with questions of understanding in the writing.

Chapter 9. Characters and group structure.
Writing down (until July 21st) for chapter 9, for independent editing, with questions of understanding in the writing.


Exercise Sheets:
Sheet 1 (until 1.5.) - first exercise sheet for chapter 1, for independent processing. Written tasks: 2 and 4

Sheet 2 (until 8.5.) - Exercise sheet on Galois theory, for independent processing. Written tasks: 3 and 4

Sheet 3 (until May 15) - Exercise sheet on Galois theory, will be worked on in groups.
Solution to exercise 2.

Sheet 4 (up to 9.6.) - Exercise sheet on polynomial equations. Written task: 2

Sheet 5 (until June 29) - Exercise sheet for modules. Written tasks: 1 and 2

Sheet 6 (up to 8 July) - Exercise sheet for modules on main ideal rings. Written tasks: 1, 3 and 5.

Sheet 7 (until July 22nd) - Exercise sheet on semi-simple algebras and characters. Written assignments: 4, 6 and 7.

Mock conditions: were announced in the Iliad and by email

Literature:
Michael Artin, algebra
Siegfried Bosch, Algebra
Gerd Fischer, textbook of algebra
Jens Carsten Jantzen and Joachim Schwermer, Algebra
Anthony Knapp, Basic Algebra
Serge Lang, algebra

The books by Bosch, Fischer and Jantzen-Schwermer are available in the university library as ebooks.

Algebra lecture in the winter semester 19/20.

Some lecture notes on algebra, some of which are also relevant for Algebra II:

(uncorrected) lecture transcript of the 2011 algebra lecture
Script for the Algebra Lecture 2017 (Prof Henke)
Algebra and Number Theory (Prof Soergel, Freiburg)
Algebra (Prof Meusburger, Erlangen)



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