TU Wien:Discrete Mathematics VO (Gittenberger)/Written Exam 2023-04-12

1) In a room there are m chairs and n people (n <= m). The people take a break and leave the room. How many ways can the n people sit on the m chairs s.t. no one sits on the same chair as before the break.

2) Let I be an integral domain. Two elements a,b of R are called associated if a=b*r with r being a unit (so element of R*). Prove that two elements x,y of R are associated if and only if x|y and y|x.

3) Let I be an integral domain. Define a~b the equivalence relation as a-b \in I. Prove that ~ is an equivalence relation. Furthermore show that [x] (which is the set of all elements related to x) is equal to x + I (or something similar to this)

4) Show that sqrt(3) + i is algebraic over Q and determine its primitive polynomial.