Chapter 9

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

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

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

Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=3+4 i \quad b=4-3 i \quad \text { in } \mathbb{Z}[i] $$

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 \(\mathbb{Z}[\sqrt{5} i]\)

In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) 13

Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=3+2 \sqrt{2} \quad b=1+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$

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. \(\begin{array}{llll}6 x-21 & \text { in } \mathbb{Z}[x], & \text { in } Q[x], & \text { in } \mathbb{Z}_{7}[x]\end{array}\)

In Exercises 5 through 8 factor the indicated Gaussian integers into a product of irreducibles in \(\mathbb{Z}[i]\) $$ -1+5 i $$

Find a quotient \(q\) and remainder \(r\) in the indicated Euclidean domain, where \(a=q b+r\). $$ a=5+2 \sqrt{2} \quad b=3+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$

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

In Exercises 8 through 11 find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=7+5 \sqrt{2} \quad b=1+\sqrt{2} \quad \text { in } \mathbb{Z}[\sqrt{2}] $$

Determine whether the indicated elements are prime in the indicated domains. If not, determine whether they are irreducible in the indicated domain. \(\begin{array}{llll}11 & \operatorname{in} \mathbb{Z}[\sqrt{3}] & \text { in } \mathbb{Z} & \text { in } \mathbb{R}\end{array}\)

Prove that there are an infinite number of primes \(p\) in \(\mathbb{Z}\) such that \(p=3\) mod 4 .

Find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=-3+7 \sqrt{3} \quad b=7-\sqrt{3} \quad \text { in } \mathbb{Z}[\sqrt{3}] $$

Determine whether the indicated elements are prime in the indicated domains. If not, determine whether they are irreducible in the indicated domain. \(\begin{array}{llll}19 & \operatorname{in} \mathbb{Z}[\sqrt{3} i] & \operatorname{in} \mathbb{Z}[\sqrt{2}] & \text { in } C\end{array}\)

(a) Prove that there are an infinite number of primes \(a+b i\) in \(\mathbb{Z}[i]\) with \(a \neq 0\) and \(b \neq 0\) (b) Prove that there are an infinite number of primes \(p\) in \(\mathbb{Z}\) such that \(p=1 \bmod 4\).

Find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=2+8 i \quad b=6+8 i \quad \text { in } \mathbb{Z}[i] $$

Find a greatest common divisor \(d\) of \(a\) and \(b\) in the indicated Euclidean domain, and express \(d=u a+v b\). $$ a=4+7 i \quad b=8-i \quad \text { in } \mathbb{Z}[i] $$

Determine whether or not the indicated integral domains are UFDs. $$ Z[x, y] $$

Let \(I\) be a nontrivial ideal in \(\mathbb{Z}[i]\). Show that \(\mathbb{Z}[i] / I\) is a finite ring.

In Exercises 12 through 14 find a generator for the ideal \(I\) in the indicated Euclidean domain. $$ I=\text { the ideal generated by } f(x)=x^{3}+x^{2}-2 x-2 \text { and } g(x)=x^{3}-x^{2}-2 x+1 \text { in } \mathrm{Q}[x] $$

Determine whether or not the indicated integral domains are UFDs. $$ \mathbb{Z}[i] $$

Let \(z\) be irreducible in \(\mathbb{Z}[i] .\) Show that \(\mathbb{Z}[i] /\langle z\rangle\) is a field.

Find a generator for the ideal \(I\) in the indicated Euclidean domain. $$ I=\text { the ideal generated by } 5+5 i \text { and } 3-i \text { in } \mathbb{Z}[i] $$

Determine whether or not the indicated integral domains are UFDs. $$ \mathbb{Z}[\sqrt{2}] $$

Find a generator for the ideal \(I\) in the indicated Euclidean domain. $$ I=\text { the ideal generated by } 13 \text { and } 3+2 i \text { in } \mathbb{Z}[i] $$

Determine whether or not the indicated integral domains are UFDs. $$ \mathbb{Z}[\sqrt{2} i] $$

In Exercises 14 through 17 find the order and characteristic of the indicated fields. $$ \mathbb{Z}[i] /\langle 7\rangle $$

Prove or disprove that \(\mathbb{Z}[x]\) is a Euclidean domain.

In Exercises 14 through 17 find the order and characteristic of the indicated fields. $$ \mathbb{Z}[i] /\langle 1+i\rangle $$

Prove or disprove that for any field \(F, F[x, y]\) is a Euclidean domain.

Show that in an integral domain \(D\), a nonzero element \(p \in D\) is prime if and only if \(\langle p\rangle\) is a prime ideal in \(D\).

Let \(D\) be a Euclidean domain and \(a\) and \(b\) elements of \(D\). Show that (a) If \(a\) and \(b\) are associates, then \(v(a)=v(b)\). (b) If \(v(a)=v(b)\) and \(a \mid b\), then \(a\) and \(b\) are associates.

Let \(D\) be a PID. Show that a nonzero element \(p \in D\) is irreducible in \(D\) if and only if \(p\) is prime in \(D\).

Let \(q\) be a prime in \(\mathbb{Z}\) such that \(q=3\) mod 4 . Show that \(\mathbb{Z}[i] /\langle q\rangle\) is a field of order \(q^{2}\)

Let \(D\) be a PID, \(p\) an irreducible element in \(D\), and \(b_{1}, \ldots, b_{r}\) elements in \(D\) such that \(p\) divides the product \(b_{1} \ldots b_{r}\). Show that \(p\) divides \(b_{i}\) for some \(1 \leq i \leq r\).

(Chinese remainder theosem) Let \(R\) be a commutative ring and \(K\) and \(L\) two proper ideals in \(R\) such that \(K+L=R\). Show that $$ R /(K \cap L) \propto R / K \times R / L $$

Let \(D\) be an integral domain. Show that the following three statements are equivalent: (a) \(D\) is a field. (b) \(D[x]\) is a Euclidean domain. (c) \(D[x]\) is a PID.

Let \(D\) be a Euclidean domain, \(a, b,\) and \(c\) nonzero elements of \(D,\) and \(d\) a greatest common divisor of \(a\) and \(b\). Show that if \(a \mid b c\), then \((a / d) \mid c\).

Let \(a\) and \(b\) be elements in a UFD. Show that if \(c\) is a ged of \(a\) and \(b\) and \(d\) is an \(\mathrm{lcm}\) of \(a\) and \(b\), then \(\mathrm{cd}\) and \(a b\) are associates.

Suppose \(x_{0}, y_{0}\) in \(\mathbb{Z}\) is a solution of the equation \(a x+b y=c,\) where \(a \neq 0, b \neq 0\), and \(c\) are in \(\mathbb{Z}\). Show that the complete set of solutions \(x, y\) in \(\mathbb{Z}\) is given by $$ x=x_{0}+k\left[b_{\operatorname{gcd}}(a, b)\right], y=y_{0}-k\left[a_{/} \operatorname{gcd}(a, b)\right] $$ for all \(k \in \mathbb{Z}\).

Let \(R\) be a commutative ring with unity. For \(a, b\) in \(R,\) a least common multiple of \(a\) and \(b\) is an element \(m \in R\) such that (1) \(a \mid m\) and \(b \mid m\) (2) If \(a \mid n\) and \(b \mid n\) for \(n \in R,\) then \(m \mid n\). Show that if \(R\) is a Euclidean domain, then (a) A least common multiple \(m\) of \(a\) and \(b\) exists. (b) Any two least common multiples of \(a\) and \(b\) are associates. (c) If \(d\) is a greatest common divisor of \(a\) and \(b\), then \(a b l d\) is a least common multiple of \(a\) and \(b\). (d) The ideal \(\langle a\rangle \cap\) \(\langle b\rangle\) is generated by any least common multiple of \(a\) and \(b\)