ABSTRACT

The structure of the groupsUn, which consist of the units in Zn, is not obvious. However, with a bit of work we can determine the structure of these groups. To do so, we will first show that the group

Up is cyclic when p is prime. We will then use this result to describe every group Un, whether n

fields and also explore splitting

When p is a prime, the ring Zp is a field. The field Zp is different than the more familiar fields like Q and R in that Zp contains only a finite number of elements. The first question we will address in this investigation is whether there are finite fields other than Zp for p prime. Activity 37.1 shows that there are in fact such fields by demonstrating the existence of a field with 4 elements. As we

will see, we can actually explicitly determine all of the finite fields.