Solution Found!
Let Zn[i] = { a+bi
Chapter 12, Problem 2CE(choose chapter or problem)
Let \(Z_n[i]\) = {\(a + bi | a, b\) belong to \(Z_n, i^2 = -1\)} (the Gaussian integers modulo \(n\) ). This software finds the group of units of this ring and the order of each element of the group. Run the program for \(n\) = 3, 7, 11, and 23. Is the group of units cyclic for these cases?
Try to guess a formula for the order of the group of units of \(Z_n[i]\) as a function of \(n\) when \(n\) is a prime and \(n\) mod 4 = 3. Run the program for \(n\) = 9 and 27. Are the groups cyclic?
Try to guess a formula for the order when \(n = 3^k\). Run the program for \(n\) = 5, 13, 17, and 29. Is the group cyclic for these cases?
What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of \(Z_n[i]\) as a function of \(n\) when n is a prime and \(n\) mod 4 = 1. Try to guess a formula for the largest order of any element in the group of units of \(Z_n[i]\) as a function of \(n\) when \(n\) is a prime and \(n\) mod 4 = 1.
On the basis of the orders of the elements of the group of units, try to guess the isomorphism class of the group. Run the program for \(n\) = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the isomorphism class of the group.
Questions & Answers
QUESTION:
Let \(Z_n[i]\) = {\(a + bi | a, b\) belong to \(Z_n, i^2 = -1\)} (the Gaussian integers modulo \(n\) ). This software finds the group of units of this ring and the order of each element of the group. Run the program for \(n\) = 3, 7, 11, and 23. Is the group of units cyclic for these cases?
Try to guess a formula for the order of the group of units of \(Z_n[i]\) as a function of \(n\) when \(n\) is a prime and \(n\) mod 4 = 3. Run the program for \(n\) = 9 and 27. Are the groups cyclic?
Try to guess a formula for the order when \(n = 3^k\). Run the program for \(n\) = 5, 13, 17, and 29. Is the group cyclic for these cases?
What is the largest order of any element in the group? Try to guess a formula for the order of the group of units of \(Z_n[i]\) as a function of \(n\) when n is a prime and \(n\) mod 4 = 1. Try to guess a formula for the largest order of any element in the group of units of \(Z_n[i]\) as a function of \(n\) when \(n\) is a prime and \(n\) mod 4 = 1.
On the basis of the orders of the elements of the group of units, try to guess the isomorphism class of the group. Run the program for \(n\) = 25. Is this group cyclic? Based on the number of elements in this group and the orders of the elements, try to guess the isomorphism class of the group.
ANSWER:Step 1 of 9
Given that \(\mathbb{Z}_{n}[i]=\left\{a+i b: a, b \in \mathbb{Z}_{n}\right\}\).
Let \(U_{n}\) be the group of units of \(\mathbb{Z}_{n}[i]\).
The result for \(n=3\) is shown below that is generated using the software.
\(\begin{array}{||l|l|} \hline \text { UNITS } & \text { ORDER } \\ \hline 0+1 \mathrm{i} & 4 \\ \hline 0+2 \mathrm{i} & 4 \\ \hline 1+0 \mathrm{i} & 1 \\ \hline 2+0 \mathrm{i} & 2 \\ \hline 1+1 \mathrm{i} & 8 \\ \hline 2+1 \mathrm{i} & 8 \\ \hline 1+2 \mathrm{i} & 8 \\ \hline 2+2 \mathrm{i} & 8 \\ \hline \end{array}\)
The total number of elements in \(U_{3}\) is 8.