This course is a fundamental course for the specialization Algebra/Number theory. In the Bachelor year, it is now possible to take this course instead of taking the course Elemente der Algebra.

An overview of the topics

Group theory

• Permutation groups
• Linear groups and representations
• Quotient groups and composition series

Fields and Galois theory

• Algebraic and transcendental field extensions
• Galois theory
• Finite fields
• Solving equations using radicals

Ring theory

• Divisibility and ideals
• Principal ideal domains and unique factorization domains
• Hilbert’s basis theorem
• Hilbert’s Nullstellensatz

Modules

• Representing finitely generated modules with generators and relations
• Finitely generated modules over principal ideal domains
• Locally free modules

Prerequisites

• Linear Algebra 1+2

 

Algebra is a 4+2 course, which can be taken to earn 9LP.

Bachelor

In the Bachelor, it is possible to take this course instead of taking Elemente der Algebra (4LP). In this case the additional 5LP will count towards the compulsory credits needed for the Wahlpflichtmodul Reine Mathematik

Master

In the Masters of Mathematik, Wirtschaftsmathematik and mathematische Biometrie you can take this lecture as as a Wahlpflichtmodul Reine Mathematik.

Lehramt

In the PO GymPO1 and in the Master Lehramt Mathematik this course can be chosen as a Wahlmodul.

Vorleistung: You will need 50% of the exercise points in order to take the exam.

Exam: There will be an oral exam.