Let D be a Euclidean domain with measure d. Prove that u

Problem 46E Find a mistake in the statement shown in Figure 18.2.

Problem 1CE Exercise 1. In the ring Z[i] (where i2 = -1) this software determines when a positive integer n is a prime. Run the program for several cases and formulate a conjecture based on your data.

Problem 1E For the ring , where d ? 1 and d is not divisible by the square of a prime, prove that the norm N(a + satisfies the four assertions made preceding Example 1. (This exercise is referred to in this chapter.)

Suppose that F is a field and there is a ring homomorphism from Z onto F. Show that F is isomorphic to \(Z_p\) for some prime p.

Problem 2E In an integral domain, show that a and b are associates if and only if

Problem 2SE Let Determine all ring automorphisms of

Show that the union of a chain \(I_{1} \subset I_{2} \subset \cdots\) of ideals of a ring R is an ideal of R. (This exercise is referred to in this chapter.)

(Second Isomorphism Theorem for Rings) Let A be a subring of R and let B be an ideal of R. Show that \(A \cap B\) is an ideal of A and that \(A/(A \cap B)\) is isomorphic to (A + B)/B. (Recall that \(\{A+B=a+b \mid a \in A, b \in B\}\).)

Problem 4E In an integral domain, show that the product of an irreducible and a unit is an irreducible.

Problem 4SE (Third Isomorphism Theorem for Rings) Let A and B be ideals of a ring R with B ? A. Show that A/B is an ideal of R/B and (R/B)/(A/B) is isomorphic to R/A.

Problem 5E Suppose that a and b belong to an integral domain, b ? 0, and a is not a unit. Show that . (This exercise is referred to in this chapter.)

Let f(x) and g(x) be irreducible polynomials over a field F. If f(x) and g(x) are not associates, prove that \(F[x] /\langle f(x) g(x)\rangle\) is isomorphic to \(F[x] /\langle f(x)\rangle \oplus F[x] /\langle g(x)\rangle\).

Problem 6E Let D be an integral domain. Define a?b if a and b are associates. Show that this defines an equivalence relation on D.

Problem 6SE (Chinese Remainder Theorem for Rings) If R is a commutative ring and I and J are two proper ideals with I + J = R, prove that R/(I ? J) is isomorphic to R/I ? R/J. Explain why Exercise 5 is a special case of this theorem.

In the notation of Example 7, show that d(xy) = d(x)d(y). EXAMPLE 7 The ring of Gaussian integers \(Z[i]=\{a+bi|a,b \in Z\}\) is a Euclidean domain with \(d(a + bi)= a^2 + b^2\). Unlike the previous two examples, in this example the function d does not obviously satisfy the necessary conditions. That \(d(x) \le d(xy)\) for \(x, y \in Z[i]\) follows directly from the fact that d(xy) = d(x)d(y).

Problem 7SE Prove that the set of all polynomials whose coefficients are all even is a prime ideal in Z[x].

Problem 8E Let D be a Euclidean domain with measure d. Prove that u is a unit in D if and only if d(u) = d(1).

Problem 8SE Let Show that I is a maximal ideal of R.

Problem 9E Let D be a Euclidean domain with measure d. Show that if a and b are associates in D, then d(a) =d(b).

Let R be a ring with unity and let a be a unit in R. Show that the mapping from R into itself given by \(x \rightarrow a x a^{-1}\) is a ring automorphism.

Problem 10E Let D be a principal ideal domain and let p ? D. Prove that is a max imal ideal in D if and only if p is irreducible.

Problem 10SE Let with b ? 0. Show that 2 does not belong to

Show that \(Z[i] /\langle 2+i\rangle\) is a field. How many elements does it have?

Problem 11E Trace through the argument given in Example 7 to find q and r in Z[i] such that 3 – 4i = (2 + 5i)q + r and d(r) < d(2 + 5i). Reference:

Problem 12E Let D be a principal ideal domain. Show that every proper ideal of D is contained in a maximal ideal of D.

Problem 12SE Is the homomorphic image of a principal ideal domain a principal ideal domain?

In \(Z[\sqrt{-5}]\), show that 21 does not factor uniquely as a product of irreducibles.

Problem 13SE For any f(x) ? Zp[x], show that f(xp) = ( f(x)) p.

Problem 14E Show that 1 –i is an irreducible in Z[i].

Problem 14SE Let p be a prime. Show that there is exactly one ring homomorphism from Zm to Zpk if pk does not divide m, and exactly two ring homomorphisms from Zm to Zpk if pk does divide m.

Show that \(Z[\sqrt{-6}]\) is not a unique factorization domain. (Hint: Factor 10 in two ways.) Why does this show that \(Z[\sqrt{-6}]\) is not a principal ideal domain?

Problem 15SE Recall that a is an idempotent if a2 = a. Show that if 1 + k is an idempotent in Zn, then n ?k is an idempotent in Zn.

Give an example of a unique factorization domain with a subdomain that does not have a unique factorization.

Problem 16SE Show that Zn (where n> 1) always has an even number of idempotents. (The number is 2d, where d is the number of distinct prime divisors of n.)

Problem 17E In Z[i], show that 3 is irreducible but 2 and 5 are not.

Problem 17SE Show that the equation x2 + y2 = 2003 has no solutions in the integers.

Prove that 7 is irreducible in \(Z[\sqrt 6]\), even though N(7) is not prime.

Prove that if both k and k + 1 are idempotents in \(Z_{n}\) and \(k \neq 0\), then n = 2k.

Problem 19E Prove that if p is a prime in Z that can be written in the form a2 + b2, then a + bi is irreducible in Z[i]. Find three primes that have this property and the corresponding irreducibles.

Problem 19SE Prove that x4 + 15x3 + 7 is irreducible over Q.

Prove that \(Z[\sqrt{-3}]\) is not a principal ideal domain.

For any integers m and n, prove that the polynomial \(x^3 + (5m + 1)x + 5n + 1\) is irreducible over Z.

In \(Z[\sqrt {-5}]\), prove that \(1 + 3 \sqrt {-5}\) is irreducible but not prime.

Prove that \(\langle\sqrt{2}\rangle\) is a maximal ideal in \(Z[\sqrt{2}]\). How many elements are in the ring \(Z[\sqrt{2}] /\langle\sqrt{2}\rangle\) ?

Problem 22E In are irreducible but not prime.

Prove that \(Z[\sqrt{-2}]\) and \(Z[\sqrt{2}]\) are unique factorization domains. (Hint: Mimic Example 7 in Chapter 18.)

Prove that \(Z [\sqrt{5}]\) is not a unique factorization domain.

Is \(\langle 3 \rangle\) a maximal ideal in Z[i]?

Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.

Problem 24SE Express both 13 and 5 + i as products of irreducibles from Z[i].

Problem 25E Let d be an integer less than –1 that is not divisible by the square of a prime. Prove that the only units of are +1 and –1.

Let \(R=\{a / b \mid a, b \in Z, 3 \times b\}\). Prove that R is an integral domain. Find its field of quotients.

Problem 26E In show that every element of the form is a unit, where n is a positive integer.

Problem 26SE Give an example of a ring that contains a subring isomorphic to Z and a subring isomorphic to Z3.

Problem 27E If a and b belong to where d is not divisible by the square of a prime and ab is a unit, prove that a and b are units.

Show that \(Z[i] /\langle 3\rangle\) is not ring-isomorphic to \(Z_{3} \oplus Z_{3}\).

For a commutative ring with unity we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show that both 2 and 3 are prime in \(Z_{12}\) but 2 is irreducible and 3 is not.

For any n > 1, prove that \(R=\left\{\left[\begin{array}{ll} a & 0 \\ 0 & b \end{array}\right] \mid a, b \in Z_{n}\right\} \) is ring-isomorphic to \(Z_n \oplus Z_n\).

Problem 29E Let n be a positive integer and p a prime that divides n. Prove that p is prime in Zn. (See Exercise 28). Reference: For a commutative ring with unity we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show that both 2 and 3 are prime in Z12 but 2 is irreducible and 3 is not.

Problem 29SE Suppose that R is a commutative ring and I is an ideal of R. Prove that R[x]/I[x] is isomorphic to (R/I)[x].

Let p be a prime divisor of a positive integer n. Prove that p is irreducible in \(Z_n\) if and only if \(p^2\) divides n. (See Exercise 28). Exercise 28 For a commutative ring with unity we may define associates, irreducibles, and primes exactly as we did for integral domains. With these definitions, show that both 2 and 3 are prime in \(Z_{12}\) but 2 is irreducible and 3 is not.

Problem 31E Prove or disprove that if D is a principal ideal domain, then D[x] is a principal ideal domain.

Problem 31SE Find an ideal I of Z8[x] such that the factor ring Z8[x]/I is an integral domain but not a field.

Problem 32E Determine the units in Z[i].

Problem 30SE Find an ideal I of Z8[x] such that the factor ring Z8[x]/I is a field.

Problem 32SE Find an ideal I of Z[x] such that Z[x]/I is ring-isomorphic to Z3

Problem 33E Let p be a prime in an integral domain. If p | a1a2 · · · an, prove that p divides some ai. (This exercise is referred to in this chapter.)

Problem 35E Let D be a principal ideal domain and p an irreducible element of D. Prove that is a field.

Problem 39E Show that for any nontrivial ideal I of Z[i], Z[i]/I is finite.

Problem 34E Show that 3x2 + 4x + 3 ? Z5[x] factors as (3x + 2)(x + 4) and (4x + 1)(2x + 3). Explain why this does not contradict the corollary of Theorem 18.3. Reference:

An ideal A of a commutative ring R with unity is said to be finitely generated if there exist elements \(a_1,a_2,\ldots a_n\) of A such that \(A= \langle a_1, a_2, \ldots ,a_n \rangle\).An integral domain R is said to satisfy the ascending chain condition if every strictly increasing chain of ideals \(I_1 \subset I_2 \subset \cdots\) must be finite in length. Show that an integral domain R satisfies the ascending chain condition if and only if every ideal of R is finitely generated.

Problem 38E Prove or disprove that a subdomain of a Euclidean domain is a Euclidean domain.

Show that an integral domain with the property that every strictly decreasing chain of ideals \(I_{1} \supset I_{2} \supset \cdots\) must be finite in length is a field.

Problem 40E Find the inverse of What is the multiplicative order of

Problem 41E In are not associates.

Problem 43E Prove that in a unique factorization domain, an element is irreducible if and only if it is prime.

Problem 42E Let R = Z ? Z ? · · · (the collection of all sequences of integers under component wise addition and multiplication). Show that R has ideals I1, I2, I3, . . . with the property that I1 ? I2 ? I3? · · · . (Thus R does not have the ascending chain condition.)

Let F be a field and let R be the integral domain in F[x] generated by \(x^2\) and \(x^3\). (That is, R is contained in every integral domain in F[x] that contains \(x^2\) and \(x^3\).) Show that R is not a unique factorization domain.

Problem 45E Prove that for every field F, there are infinitely many irreducible elements in F[x].