Integral Domains: exercises

Practice (19)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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 \(-1\).

   Mutasd meg a választ!

   Hint:

   Note that \(z^{-1} = 1/(a + b\sqrt{3}\, i) = (a -b \sqrt{3}\, i)/(a^2 + 3b^2)\) is in \({\mathbb Z}[\sqrt{3}\, i]\) if and only if \(a^2 + 3 b^2 = 1\). The only integer solutions to the equation are \(a = \pm 1, b = 0\).

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

   1. \(5\)

   2. \(1 + 3i\)

   3. \(6 + 8i\)

   4. \(2\)

   Mutasd meg a választ!

   Hint:

   (a) \(5 = -i(1 + 2i)(2 + i)\); (c) \(6 + 8i = -i(1 + i)^2(2 + i)^2\).

3. Let \(D\) be an integral domain.

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

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

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

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

   Mutasd meg a választ!

   Hint:

   True.

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

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

7. Let \(F\) be a field.

   1. 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. \(F(x)\) field of rational functions in \(x\)

   2. 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 \(F(x_1, \ldots, x_n)\). \(F(x_1, \dots, x_n)\) field of rational functions in \(x_1, \ldots, x_n\)

8. 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 \(p\).

9. Prove that the field of fractions of the Gaussian integers, \({\mathbb Z}[i]\), is \[\begin{aligned}\end{aligned}\].

   Mutasd meg a választ!

   Hint:

   Let \(z = a + bi\) and \(w = c + di \neq 0\) be in \({\mathbb Z}[i]\). Prove that \(z/w \in {\mathbb Q}(i)\).

10. 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 \(F\).

    1. Prove that every field contains a unique prime subfield.

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

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

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

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

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

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

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

12. 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 \(b\).

    1. 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 \(b\).

    2. 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 \(\gcd(a, b) = as + bt\).

13. 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 \(D\).

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

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

    Mutasd meg a választ!

    Hint:

    Let \(a = ub\) with \(u\) a unit. Then \(\nu(b) \leq \nu(ub) \leq \nu(a)\). Similarly, \(\nu(a) \leq \nu(b)\).

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

    Mutasd meg a választ!

    Hint:

    Show that \(21\) can be factored in two different ways.

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

18. 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 \(R\). Prove that \(R\) satisfies the ascending chain condition if and only if every ideal of \(R\) is finitely generated.

19. 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.

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

i

imaginary unit
i² = −1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

Questions people ask

What is a group, in plain words?

A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group — that is where the idea came from.

What is the difference between a ring and a field?

A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.

Próbáld a sajátodat.

⌨ Type ✎ Draw

A matematikai problémád

Megoldás

x²√÷×π( )=∫d/dxlim[ ]sinlogeˣ

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

Még több Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula