91Ó°ÊÓ

Determine all the units in the indicated domains: (a) \(\mathbb{Z}[\sqrt{2} i]\) (b) \(\mathbb{Z}_{7}[x]\) (c) \(\mathbb{Z}[i][x]\) (d) \(C[x]\)

In Exercises 1 through 4 express the indicated primes in \(\mathbb{Z}\) as the sum of two squares and write their factorizations into irreducibles in \(\mathbb{Z}[i]\). 17

In Exercises 1 through 4 express the indicated primes in \(\mathbb{Z}\) as the sum of two squares and write their factorizations into irreducibles in \(\mathbb{Z}[i]\). 29

In Exercises 2 through 6 determine whether the indicated pairs of elements are associates in the indicated domains. 1 and \(2+\sqrt{3}\) in \(\mathbb{Z}[\sqrt{3}]\)

Determine whether the indicated pairs of elements are associates in the indicated domains. \(\begin{array}{lll}5 x-10 & \text { and } x-2 & \text { in } \mathbb{Z}[x]\end{array}\)

In Exercises 1 through 4 express the indicated primes in \(\mathbb{Z}\) as the sum of two squares and write their factorizations into irreducibles in \(\mathbb{Z}[i]\). 37

Determine whether the indicated pairs of elements are associates in the indicated domains. \(\begin{array}{lll}3+2 \sqrt{2} & \text { and } 1-\sqrt{2} & \text { in } \mathbb{Z}[\sqrt{2}]\end{array}\)

In Exercises 4 through 7 find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=5+3 i \quad b=2+i \quad \text { in } \mathbb{Z}[i] $$

In Exercises 1 through 4 express the indicated primes in \(\mathbb{Z}\) as the sum of two squares and write their factorizations into irreducibles in \(\mathbb{Z}[i]\). 41

In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) 11