Algebra - 2020
Vivek, 17.08., 14:30 - 15:00
Mr. Steinmann let me in at 14:32 and the exam began at 14:35. Wearing a mask is optional and you can either write on a big black board or just answer out loud, Prof. Pink does not care. After overhearing me talking to myself in english Prof. Pink offered to conduct the exam in English. so I believe you also have to option to do the exam in English. He is very chill.
Algebra I:
- Definition of an integral domain, definition of a prime ideal. State and prove an equivalent definition of a prime ideal, namely: ideal \(I\) is prime iff \(A/I\) is an integral domain. (only had to show one direction (left to right))
- Give examples of prime- and non-prime ideals.
- Definition of a quotient field, the quotient field of \(\mathbb{Z}[i]\) and prove it is (isomorphic to) \(\mathbb{Q}(i)\)
- Definition of \(S_n\) and state all properties you know. (\(|S_n|=n!\), all finite groups are isomorphic to a subgroup of \(S_n\), for \(n\geq 5\), the only normal subgroups of \(S_n\) are \(S_n\), \(A_n\) and the trivial subgroup, \(S_n\) is solvable iff \(n\leq 4\)).
- Definition of a solvable group. Prove \(S_4\) is solvable. (My brain went brrr here and I blurted out the klein group is normal in \(S_4\) lol )
- Prove \(S_4/K_4\) is isomorphic to \(S_3\). (apparently we did this in lecture and even Mr. Steinmann was surprised, then Prof. Pink said "Ah, da hat jetzt auch Herr Steinmann etwas neues dazu gelernt", the statement is in page 45 btw)
Algebra II:
- Definition of algebraic closure, when does one exist? Idea of the proof.
- The algebraic closure of \(\mathbb{Q}\).
- Galois group of \(X^7+1\). (Again; brain went *windows shutdown sound* and could not see it is separable just like \(X^7-1\) but he let me work with \(X^7-1\) and its galois group is of course \((\mathbb{Z}/7\mathbb{Z})^\times\))
- Subfields of the Galois group. Construct them (I ran out of time here)
They spent 22 minutes on Algebra I and 8 minutes on Algebra II. They let you speak and you will(!) get interrupted if you make a tiny mistake. Prof. Pink also gives useful hints sometimes and if he sees that you understand the concept, he is nice enough to stop you and skips it. In general both of them are very friendly but I was still quite nervous. If you make mistakes like I did (klein group is normal in \(S_4\) they guide you through it and allow you to correct yourself, which I found very nice. I recommend talking as fast as you can, Prof. Pink is a beast and understands everything you say at any rate.
