Difference between revisions of "Algebra - 2018"

Revision as of 10:02, 6 August 2018

Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like me kicking you with my fists.

Laurena Python, 06.08.2018, 09:30-10:00

Algebra I:

• Definition of integral domain, examples, Z/pZ is an integral domain if and only p is prime (with proof)
• What is a fraction field ? Sketch the construction
• Definition of prime and maximal ideal, examples, give a prime/maximal ideal of C[X,Y]
• State CRT and give a proof idea, definition coprime ideals
• Statement of 1st isomorphism theorem for rings

Algebra II:

• Definition of the Galois group of a polynomial, definition of the splitting field
• Definition of separablility (for a polynomial)
• What is the cardinality of the Galois group of a separable polynomial ?
• What can we say about the cardinality of the Galois group of an irreducible separable polynomial ? Why ?
• Definition of "solvable by radicals", definition of radical extension, definition of pure extension
• Galois group of X^3-2
• Galois group of F_p^n/F_p

They spend 20 minutes on Algebra I (Kowalski asks the questions) and 10 minutes on Algebra II (Burger asks the questions). They let you speak and do not interrupt, they also give useful hints. The proofs do not have to be written formally. If they see that you understand the concept, they sometimes stop you before the end.

Felix Sefzig, 06.08.2018, 11:00-11:30

Algebra I:

• Definition of integral domain with examples
• What is special about K[X]? Euclidean division and PID
• Find maximal & prime but not maximal ideals of C[X,Y,Z]
• What is conjugation & conjugacy classes
• Conjugacy classes in GL(n,C)
• Definition of group action, G-morphism and examples
• Orbit-Stabelizer-Theorem

Algebra II:

• Definiton of Galois group, what can you say about the cardinality
• Galois group of F_{p^n}/F_{p}
• f separable iff gcd(f,f') = 1
• Normal extension, Galois extension