18.4: Exercises
Let \(z = a + b \sqrt{3}\, i\) be in \({\mathbb Z}[ \sqrt{3}\, i]\text{.}\) If \(a^2 + 3 b^2 = 1\text{,}\) show that \(z\) must be a unit. Show that the only units of \({\mathbb Z}[ \sqrt{3}\, i ]\) are \(1\) and \(-1\text{.}\)
The Gaussian integers, \({\mathbb Z}[i]\text{,}\) are a UFD . Factor each of the following elements in \({\mathbb Z}[i]\) into a product of irreducibles.
- \(\displaystyle 5\)
- \(\displaystyle 1 + 3i\)
- \(\displaystyle 6 + 8i\)
- \(\displaystyle 2\)
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, \(F_D\text{.}\)
- Verify the associative and commutative properties for multiplication in \(F_D\text{.}\)
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 \(D\text{.}\)
Let \(F\) be a field of characteristic zero. Prove that \(F\) contains a subfield isomorphic to \({\mathbb Q}\text{.}\)
Let \(F\) be a field.
- Prove that the field of fractions of \(F[x]\text{,}\) denoted by \(F(x)\text{,}\) is isomorphic to the set all rational expressions \(p(x) / q(x)\text{,}\) 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 \(F[x_1, \ldots, x_n]\text{.}\) 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 \(F[x_1, \ldots, x_n]\text{.}\) We denote the field of fractions of \(F[x_1, \ldots, x_n]\) by \(F(x_1, \ldots, x_n)\text{.}\)
Let \(p\) be prime and denote the field of fractions of \({\mathbb Z}_p[x]\) by \({\mathbb Z}_p(x)\text{.}\) Prove that \({\mathbb Z}_p(x)\) is an infinite field of characteristic \(p\text{.}\)
Prove that the field of fractions of the Gaussian integers, \({\mathbb Z}[i]\text{,}\) is
\[ {\mathbb Q}(i) = \{ p + q i : p, q \in {\mathbb Q} \}\text{.} \nonumber \]
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 \(F\text{.}\)
- 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, \({\mathbb Q}\text{.}\)
- If \(F\) is a field of characteristic \(p\text{,}\) prove that the prime subfield of \(F\) is isomorphic to \({\mathbb Z}_p\text{.}\)
Let \({\mathbb Z}[ \sqrt{2}\, ] = \{ a + b \sqrt{2} : a, b \in {\mathbb Z} \}\text{.}\)
- Prove that \({\mathbb Z}[ \sqrt{2}\, ]\) is an integral domain.
- Find all of the units in \({\mathbb Z}[\sqrt{2}\, ]\text{.}\)
- Determine the field of fractions of \({\mathbb Z}[ \sqrt{2}\, ]\text{.}\)
- Prove that \({\mathbb Z}[ \sqrt{2} i ]\) is a Euclidean domain under the Euclidean valuation \(\nu( a + b \sqrt{2}\, i) = a^2 + 2b^2\text{.}\)
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 \(b\text{.}\)
- If \(D\) is a PID and \(a\) and \(b\) are both nonzero elements of \(D\text{,}\) 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 \(b\text{,}\) then \(d\) and \(d'\) are associates. We write \(\gcd( a, b)\) for the greatest common divisor of \(a\) and \(b\text{.}\)
- Let \(D\) be a PID and \(a\) and \(b\) be nonzero elements of \(D\text{.}\) Prove that there exist elements \(s\) and \(t\) in \(D\) such that \(\gcd(a, b) = as + bt\text{.}\)
Let \(D\) be an integral domain. Define a relation on \(D\) by \(a \sim b\) if \(a\) and \(b\) are associates in \(D\text{.}\) Prove that \(\sim\) is an equivalence relation on \(D\text{.}\)
Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\text{.}\) If \(u\) is a unit in \(D\text{,}\) show that \(\nu(u) = \nu(1)\text{.}\)
Let \(D\) be a Euclidean domain with Euclidean valuation \(\nu\text{.}\) If \(a\) and \(b\) are associates in \(D\text{,}\) prove that \(\nu(a) = \nu(b)\text{.}\)
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 the ideal can be written as \(a_1 r_1 + \cdots + a_n r_n\) for some \(r_1, \ldots, r_n\) in \(R\text{.}\) 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 \(I_1 \supset I_2 \supset I_3 \supset \cdots\text{.}\) Suppose that there exists an \(N\) such that \(I_k = I_N\) for all \(k \geq N\text{.}\) 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 \(a, b \in S\text{.}\)
- Define a relation \(\sim\) on \(R \times S\) by \((a, s) \sim (a', s')\) if there exists an \(s^\ast \in S\) such that \(s^\ast(s' a -s a') = 0\text{.}\) Show that \(\sim\) is an equivalence relation on \(R \times S\text{.}\)
-
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 \(\sim\text{.}\) 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}\text{,} \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 \(S\text{.}\)
- 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 \(0 \notin S\text{,}\) 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 \(R\text{.}\)
- If \(P\) is a prime ideal of \(R\) and \(S = R \setminus P\text{,}\) 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 .