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Let \(z = a + b \sqrt{3}\, i\) be in \({\mathbb Z}[ \sqrt{3}\, i]\). If \(a^2 + 3 b^2 = 1\), show that \(z\) must be a unit. Show that the only units of \({\mathbb Z}[ \sqrt{3}\, i ]\) are 1 and \(-1\).

Let \(z = a + b \sqrt{3}\, i\) be in \({\mathbb Z}[ \sqrt{3}\, i]\). If \(a^2 + 3 b^2 = 1\), show that \(z\) must be a unit. Show that the only units of \({\mathbb Z}[ \sqrt{3}\, i ]\) are 1 and \(-1\).

The Gaussian integers, \({\mathbb Z}[i]\), are a UFD. Factor each of the following elements in \({\mathbb Z}[i]\) into a product of irreducibles.

5

\(1 + 3i\)

\(6 + 8i\)

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Let \(D\) be an integral domain.

Prove that \(F_D\) is an abelian group under the operation of addition.

Show that the operation of multiplication is well-defined in the field of fractions, \(F_D\).

Verify the associative and commutative properties for multiplication in \(F_D\).

Prove or disprove: Any subring of a field \(F\) containing 1 is an integral domain.

Prove or disprove: If \(D\) is an integral domain, then every prime element in \(D\) is also irreducible in \(D\).

Let \(F\) be a field of characteristic zero. Prove that \(F\) contains a subfield isomorphic to \({\mathbb Q}\).

Let \(F\) be a field.

Prove that the field of fractions of \(F[x]\), denoted by \(F(x)\), is isomorphic to the set all rational expressions \(p(x) / q(x)\), where \(q(x)\) is not the zero polynomial.

Let \(p(x_1, \ldots, x_n)\) and \(q(x_1, \ldots, x_n)\) be polynomials in \(F[x_1, \ldots, x_n]\). Show that the set of all rational expressions \(p(x_1, \ldots, x_n) / q(x_1, \ldots, x_n)\) is isomorphic to the field of fractions of \(F[x_1, \ldots, x_n]\). We denote the field of fractions of \(F[x_1, \ldots, x_n]\) by \(F(x_1, \ldots, x_n)\).

Let \(p\) be prime and denote the field of fractions of \({\mathbb Z}_p[x]\) by \({\mathbb Z}_p(x)\). Prove that \({\mathbb Z}_p(x)\) is an infinite field of characteristic \(p\).

Prove that the field of fractions of the Gaussian integers, \({\mathbb Z}[i]\), is \begin{equation*}{\mathbb Q}(i) = \{ p + q i : p, q \in {\mathbb Q} \}.\end{equation*}

A field \(F\) is called a *prime field* if it has no proper subfields. If \(E\) is a subfield of \(F\) and \(E\) is a prime field, then \(E\) is a *prime subfield* of \(F\).

Prove that every field contains a unique prime subfield.

If \(F\) is a field of characteristic 0, prove that the prime subfield of \(F\) is isomorphic to the field of rational numbers, \({\mathbb Q}\).

If \(F\) is a field of characteristic \(p\), prove that the prime subfield of \(F\) is isomorphic to \({\mathbb Z}_p\).

Let \({\mathbb Z}[ \sqrt{2}\, ] = \{ a + b \sqrt{2} : a, b \in {\mathbb Z} \}\).

Prove that \({\mathbb Z}[ \sqrt{2}\, ]\) is an integral domain.

Find all of the units in \({\mathbb Z}[\sqrt{2}\, ]\).

Determine the field of fractions of \({\mathbb Z}[ \sqrt{2}\, ]\).

Prove that \({\mathbb Z}[ \sqrt{2} i ]\) is a Euclidean domain under the Euclidean valuation \(\nu( a + b \sqrt{2}\, i) = a^2 + 2b^2\).

Let \(D\) be a UFD. An element \(d \in D\) is a *greatest common divisor of \(a\) and \(b\) in \(D\)* if \(d \mid a\) and \(d \mid b\) and \(d\) is divisible by any other element dividing both \(a\) and \(b\).

If \(D\) is a PID and \(a\) and \(b\) are both nonzero elements of \(D\), prove there exists a unique greatest common divisor of \(a\) and \(b\) up to associates. That is, if \(d\) and \(d'\) are both greatest common divisors of \(a\) and \(b\), then \(d\) and \(d'\) are associates. We write \(\gcd( a, b)\) for the greatest common divisor of \(a\) and \(b\).

Let \(D\) be a PID and \(a\) and \(b\) be nonzero elements of \(D\). Prove that there exist elements \(s\) and \(t\) in \(D\) such that \(\gcd(a, b) = as + bt\).

Let \(D\) be an integral domain. Define a relation on \(D\) by \(a \sim b\) if \(a\) and \(b\) are associates in \(D\). Prove that \(\sim\) is an equivalence relation on \(D\).

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\). If \(u\) is a unit in \(D\), show that \(\nu(u) = \nu(1)\).

Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\). If \(a\) and \(b\) are associates in \(D\), prove that \(\nu(a) = \nu(b)\).

Show that \({\mathbb Z}[\sqrt{5}\, i]\) is not a unique factorization domain.

Prove or disprove: Every subdomain of a UFD is also a UFD.

An ideal of a commutative ring \(R\) is said to be *finitely generated* if there exist elements \(a_1, \ldots, a_n\) in \(R\) such that every element \(r \in R\) can be written as \(a_1 r_1 + \cdots + a_n r_n\) for some \(r_1, \ldots, r_n\) in \(R\). Prove that \(R\) satisfies the ascending chain condition if and only if every ideal of \(R\) is finitely generated.

Let \(D\) be an integral domain with a descending chain of ideals \(I_1 \supset I_2 \supset I_3 \supset \cdots\). Suppose that there exists an \(N\) such that \(I_k = I_N\) for all \(k \geq N\). A ring satisfying this condition is said to satisfy the *descending chain condition*, or *DCC*. Rings satisfying the DCC are called *Artinian rings*, after Emil Artin. Show that if \(D\) satisfies the descending chain condition, it must satisfy the ascending chain condition.

Let \(R\) be a commutative ring with identity. We define a *multiplicative subset* of \(R\) to be a subset \(S\) such that \(1 \in S\) and \(ab \in S\) if \(a, b \in S\).

Define a relation \(\sim\) on \(R \times S\) by \((a, s) \sim (a', s')\) if there exists an \(s^\ast \in S\) such that \(s^\ast(s' a -s a') =0\). Show that \(\sim\) is an equivalence relation on \(R \times S\).

Let \(a/s\) denote the equivalence class of \((a,s) \in R \times S\) and let \( S^{-1}R\) be the set of all equivalence classes with respect to \(\sim\). Define the operations of addition and multiplication on \(S^{-1} R\) by \begin{align*} \frac{a}{s} + \frac{b}{t} & = \frac{at + b s}{s t}\\ \frac{a}{s} \frac{b}{t} & = \frac{a b}{s t}, \end{align*} respectively. Prove that these operations are well-defined on \(S^{-1}R\) and that \(S^{-1}R\) is a ring with identity under these operations. The ring \(S^{-1}R\) is called the

*ring of quotients*of \(R\) with respect to \(S\).Show that the map \(\psi : R \rightarrow S^{-1}R\) defined by \(\psi(a) = a/1\) is a ring homomorphism.

If \(R\) has no zero divisors and \(0 \notin S\), show that \(\psi\) is one-to-one.

Prove that \(P\) is a prime ideal of \(R\) if and only if \(S = R \setminus P\) is a multiplicative subset of \(R\).

If \(P\) is a prime ideal of \(R\) and \(S = R \setminus P\), show that the ring of quotients \(S^{-1}R\) has a unique maximal ideal. Any ring that has a unique maximal ideal is called a

*local ring*.