Algebra - 2018
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Contents
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, connection of Galois groups in tower of normal extensions
- Galois extension
Lukas, 06.08.2018, 13:30-14:00
Algebra I
- Groups: Examples and Homomorphisms between them
- GL(n,R) and the subgroups -> Special Linear Group SL(n,R) (and show that it's normal)
- for n=2; Non-normal subgroups of GL(n,R) (triangular matrices)
- Center and Conjugacy Class (and the connection Normal and Conjugacy class)
- Lagrange Theorem with proof, and application (Ord(x) | card(G))
- What is special subgroups of index 2? (they are normal, with proof)
- Does the converse of Lagrange hold too? (No, and no proof asked)
Algebra II
- Definition of Splitting field of Polynomial
- Gal(Fp^n : Fp) is isomorphic to Z/nZ, with proof
- Def of solvable by radicals
Manuel, 06.08.2018, 15:30-16:00
Algebra I
- polynomial ring of a field: Polynomial division & Proof of why it works out
- K[X] is PID unsing the proven statment above
- ring: Every PID is a UFD
- definition UFD
- groups: orbit-stabilizer theorem
- definition group action with example
Algebra II
- definition Galois group
- definition normal
- irreducible polynomial degree n, n divides cardinality Galois group
- three equivalences Galois extension
