TU Wien:Diskrete Mathematik für Informatik VO (Gittenberger)/Schriftliche Prüfung 09.05.2014

Task 1

Show that $\mathbb {Z} [i]=\{a+bi\mid a,b\in \mathbb {Z} \}$ is a subring of $(\mathbb {C} ,+,.)$

Find all units.

Is $\mathbb {Z} [i]$ an integral domain?

Task2

Is ${\sqrt {6}}+2{\sqrt {3}}$ algebraic over $\mathbb {Q}$?

Find minimal polynom.

Task 3

Solve two systems of modulo congruences (forgot the exact numbers)

(1) $2x\equiv 4(8),.....$

(2) $4x\equiv 2(8),.....$

Task 4

$M=\{1..n\}$

Graph $G=(V,E)$ where

$V(G)=P(M)=\{U\mid U\in P(M)\}$

$E(G)=\{U,V\mid U,V\in V(G),U\neq V,U\cap V=\emptyset \}$

Calculate $\alpha _{0}(G){\text{ and }}\alpha _{1}(G)$