TU Wien:Discrete Mathematics VO (Gittenberger)/Written Exam 2024-11-08

Exersice 1

Is $\mathbb {Z} _{5}[x]/x^{4}+x^{2}+1$ a field ? List all members of it. Is $x+2$ a unit in it? If so, find its multiplicative inverse.

Exersice 2

Determine the number of spanning trees for a given graph (consisting of two connected components and 8 total vertices) using the matrix-tree-theorem.

Exersice 3

Let $\varphi$ be a ring homomorphism and $R,S$ two rings. Let $I$ be an ideal of $S$. Proof that $\varphi ^{-1}=\{x\in R\mid \varphi (x)\in I\}$ is an ideal as well.

Exersice 4

A Hasse diagram of a post was given. Determine the value of the Möbius function $\mu (0,1)$.