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Magazine Problem 5
Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=3+4 i \quad b=4-3 i \quad \text { in } \mathbb{Z}[i] $$
Problem 6
Determine whether the indicated pairs of elements are associates in the indicated domains. \(1+\sqrt{5} i \quad\) and \(1-\sqrt{5} i \quad\) in \(\mathbb{Z}[\sqrt{5} i]\)
Problem 6
In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) 13
Problem 6
Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=3+2 \sqrt{2} \quad b=1+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$
Problem 7
In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) $$ -1+5 i $$
Problem 7
Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=5+2 \sqrt{2} \quad b=3+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$
Problem 7
In Exercises 7 through 10 determine whether the indicated elements are prime in the indicated domains. If not, determine whether they are irreducible in the indicated domain. \(\begin{array}{llll}6 x-21 & \text { in } \mathbb{Z}[x], & \text { in } Q[x], & \text { in } \mathbb{Z}_{7}[x]\end{array}\)
Problem 8
In Exercises 8 through 11 find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=7+5 \sqrt{2} \quad b=1+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$
Problem 8
In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) $$ 8-i $$
Problem 9
Find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=-3+7 \sqrt{3} \quad b=7-\sqrt{3} \quad \text { in } \mathbb{Z}[\sqrt{3}] $$
