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
- Rings
- Polynomial ring of a field: Polynomial division & Proof of why it works out
- Proof that \( K[X] \) is a PID unsing the proven statment above
- Definition UFD
- Every PID is a UFD with Proof
- Groups
- Definition of a group action with an example where \( G \) acts on itself
- Orbit-stabilizer theorem + Start of proof (no time left)
Algebra II
- Definition Galois group of a Polynomial
- Definition Splitting field
- Definition Normal extension
- Wierd question about what i can say about the cardinalitiy of tha Galois group eventiully leading to...
- Given an irreducible polynomial of degree \( n \), i had to show that \( n\) divides the cardinality of the Galois group
- Define the three equivalences of a Galois extension (He wanted me to proove one direction, i think that the second one implies the third one but time didn't suffice)
All in all it seemed rather formal to me, K. as always is more straight forward and in my case more demanding, i guess, than B. which sometimes glances at K. with a smear grin when you say something funny in his opinion - made me is a bit unsetteled(perhaps just in my case bc he went for a cig break before). The exam flows exactly the way K. wants it in his part but it seems more free in B.'s Part.
Joël, 07.08.2018, 08:30-09:00
Algebra I:
- Show that \( \mathbb{Z}/m\mathbb{Z} \) is an integral domain iff \(p\) is prime. At one point I considered a prime decomposition \(m = p_1\ldots p_n\) and continued with the proof, whereupon he interrupted me and said that I must state \(n \geq 2\).
- Definition of the characteristic of a field, show that it must be prime.
- Definition of the Frobenius automorphism and show that it is a field morphism, ie show \(p \nmid \)\({p}\choose{k} \) for \(k \neq 1, \; k \neq p\)
- Definition of polynomial rings as finite sequences in \(R\) with definition of addition, product, and which sequence corresponds to \(X\).
- Show that if \(R\) is an integral domain, then \(R[X]\) is an integral domain.
Algebra II:
- Definition of the Galois group of a polynomial, definition of the splitting field, why does it exist? Answer is Kronecker's theorem (without proof), which asserts that for every \(f \in k[X]\) there is a field \(L\) in which \(f\) splits, and then \(R(f) \subset L, \; k \subset k(R(f)) \subset L\) is a splitting field.
- Relation between the Galois group and the set of roots.
- The Galois group injects into \(S_{R(f)}\); cases where the Galois group is not isomorphic to \(S_{R(f)}\); Give an example of a polynomial whose Galois group is isomorphic to \(S_2\), I suggested \(X^2 + 1 \in \mathbb{Q}[X]\).
- Definition of a separable polynomial, \(|Gal(E/k)| = [E:k]\) in that case (without proof).
- Show that \(Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) \cong \mathbb{Z}/n\mathbb{Z}\).
- Definition of a normal extension, characterisation of Galois extensions, without proof.
