# Algebra Questions Please help

## algebra

##### Mathematical Institute of the Heinrich Heine University Düsseldorf

### Karin Halupczok

HHU Düsseldorf, summer semester 2021**Lecturer: PD Dr. Karin Halupczok Assistant: Dr. David Bradley-Williams**

### Dates of the lecture:

**Mon 08:30 - 10:00**

**Wed 08:30 - 10:00**

Link in the LSF. The lecture will be held online as a live video conference. The access link is visibly announced there for those registered in the LSF. You can also find the link in the Iliad system.

### Lecture notes (handwritten):

Script: A0A1A2A3A4A5A6A7A8A9A10A11A12Lecture notes: Notes_A1Notizen_A2Notizen_A3Notizen_A4Notizen_A5Notizen_A6Notizen_A7Notizen_A8Notizen_A9Notizen_A10Notizen_A11

### Organization of the exercises:

Link to the algebra exercises in the LSF. Please register there via the LSF for the exercise group you want (ideally by April 17th). Please contact Dr. Email David Bradley-Williams or Rocketchat if you're having trouble registering on the LSF.The following exercise dates are offered (exercise starts in the week of April 19): The exercises will be held online as a live video conference via Webex. The Webex links can be found on the ILIAS Exercises on Algebra page (you may have to click twice, the first time to log into ILIAS).

Please join the Rocketchat channel # Algebra-2021, where you can discuss and ask questions with the tutor and your fellow students.

**Additional exercises because of Pentecost**: Of course on Whit Monday there are no classes for ÜB1 and ÜB2, instead the following two replacements (with the same Webex links as always):

### Submission of exercise sheets:

Under "Exercises in Algebra" in ILIAS, you can regularly upload your solutions as a PDF file to the relevant sheet.The deadline for each

**Wednesdays before 12:10**Be clock.

Please note the following rules for submitting solutions.

- Your "Team" submission must be size 1 or 2.
- Submissions from a team greater than 2 receive 0 points.
- Copied files that are submitted to different "teams" are rated and determined with 0.
- All submissions should be uploaded as a single PDF files.
- Upload your solution file early because you have tried a few exercises. You or your partner can upload additional versions by the deadline.
- Check that your final submission file is the one you want by downloading it from Iliad before the deadline. Even if your partner: has uploaded to the file.
- All parts of your solution that are illegible / fuzzy must be ignored by the corrector and you can get 0 points for this task.
- Please then delete all old versions (except for your desired edition version)
**in front**the deadline.

**now**in the rocket chat group for advice. Of course, you can also ask in the Rocket chat if you are looking for a partner for the team handover.

Questions / problems about the submission (as well as math questions!) Are best asked in the Rocket chat group. There are lots of people there who can help (probably faster than email), and the answers can help everyone!

### Exercises:

- Sheet 1 (submission by ILIAS, deadline April 21, 21, by 12:10 pm) [suggested solutions]
- Sheet 2 (submission by ILIAS, deadline April 28, 21, by 12:10 pm) [suggested solutions]
- Sheet 3 (submission by ILIAS, deadline 05.05.21, by 12:10 pm) [suggested solutions]
- Sheet 4 (submission by ILIAS, deadline May 12th, 21, by 12:10 pm) [suggested solutions]
- Sheet 5 (submission by ILIAS, deadline 19.05.21, by 12:10 pm) [suggested solutions]
- Sheet 6 (submission by ILIAS, deadline May 26th, 21, by 12:10 p.m.)

### Exam:

In order to pass the lecture module, you have to pass an exam at the end of the lecture.Those who have achieved 40% of the exercise points or who have previously unsuccessfully participated in an algebra exam are admitted to the exam. Physics and computer science students can take the written exam even if they have previously been admitted to the algebra exam.

The

**Exam dates**will be announced here as soon as they are known.

### Lecture commentary:

In the lecture the basic algebraic structures in mathematics are dealt with in detail, i.e. groups, rings and solids. The aim is to understand the solutions of polynomial equations in an indeterminate. These roots are usually in a finite extension of the body. Such body extensions can mostly be written as the quotient of the polynomial ring after an irreducible polynomial. If several roots appear, these are permuted by a group of automorphisms, which leads to the so-called Galois theory. In the lecture the necessary basic terms for groups, rings and bodies are developed. In the end we will see, as an application of Galois theory, that the roots of a general equation of degree at least five can no longer be expressed by a solution formula that is analogous to the formula for quadratic equations.

### Contents of the lecture in key words:

- A0: Introduction and sentences in Z

- A1: Groups: Definitions and Examples
- A2: subgroups, factor groups
- A3: group operations
- A4: group homomorphisms
- A5: The symmetrical groups S_n
- A6: Direct sums of Abelian groups
- A7: Finally generated Abelian groups
- A8: Solvable groups
- A9: Sylow groups

- A10: Rings: Definitions and Examples
- A11: Ideals, factor rings, homomorphisms
- A12: Commutative Rings, Integrity Regions
- A13: Main ideal areas
- A14: polynomial rings
- A15: Primitive and irreducible polynomials
- A16: Modules: Definitions and Examples
- A17: Finally generated modules over main ideal areas

- A18: Algebraic extensions
- A19: disintegration bodies
- A20: separability
- A21: Finite bodies

- A22: Galois groups
- A23: Main theorem of Galois theory
- A24: radical extensions
- A25: Radical dissolvability
- A26: Constructions with compasses and ruler

### Literature:

(is only a small selection of many suitable books)- Fischer, G .: Textbook of Algebra
- Artin, M .: Algebra
- Bosch, S .: Algebra
- Lang, S .: Algebra

Last change: May 21, 2021 (DBW)

Responsible for the content: Karin Halupczok ♦ Imprint ♦ Data protection ♦ Contact

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