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Übungsaufgaben zur Algebra 2011/2012 - Blatt 1

Abgabetermin: Do 20. Oktober 2011 am Ende der Vorlesung

For integers \(m\ge 1\) and \(x\), we use in the following \([x]_m\) for the residue class modulo \(m\) of \(x\).

Let \(m\) be a divisor of \(n\). Show that the map \(\phi:\mathbb{Z}/n\mathbb{Z}\mapsto\mathbb{Z}/m\mathbb{Z}\) given by \([x]_n\mapsto [x]_m\) is well-defined.

Find the inverse and the order of each of the primitive residue classes modulo \(24\).

The integer \(x\) is called a primitive root modulo \(n\), if for every \(y\) in \(\mathbb{Z}\) with \(\gcd(y,n)=1\), the residue class \([y]_n\) equals a power of the residue class \([x]_n\). Show that \([2]_{13}\) is a primitive root modulo \(13\). What is the order of \([2]_{13}\) modulo \(13\)?

Find integers \(x\) and \(y\) such that \(6025x+1206y=1\).

Calculate the multiplication table of the elements in \(\mathbb{Z}/12\mathbb{Z}\).