EXERCISES

EXERCISES

For Exercises 9.1 to 9.6 calculate the quotients and remainders.

For Exercises 9.7 to 9.13, find the greatest common divisors of the elements a, b in the given euclidean ring, and find elements s, t in the ring so that as + bt = gcd(a, b).

9.7.a = 33, b = 42 in Z.

9.8.a = 2891, b = 1589 in Z.

9.9.a = 2x3 − 4x2 − 8x + 1, b = 2x3 − 5x2 − 5x + 2 Q[x].

9.10.a = x6 − x3 − 16x2 + 12x − 2, b = x5 − 2x2 − 16x + 8 Q[x].

9.11.a = x4 + x + 1, b = x3 + x2 + x Z3[x].

9.12.a = x4 + 2, b = x3 + 3 Z5[x].

9.13.a = 4 − i, b = 1 + i Z[i].

For Exercises 9.14 to 9.17, find one solution to each equation with x, y Z.

For Exercises 9.18 to 9.21, find the inverse to the element in the given field.

Find all integral solutions to the equations in Exercises 9.22 to 9.24.

Factor the polynomials in Exercises 9.25 to 9.36 into irreducible factors in the given ring.

9.37. Find all irreducible polynomials of degree 5 over Z2.

9.38.Find an irreducible polynomial of degree 2 over Z5.

9.39.Find an irreducible polynomial of degree 3 over Z7.

9.40.Find the kernel and image of√the ring morphism ψ: R[x] → C defined by

=p(i), where i = −1.

9.41. Find the kernel and√image of the ring morphism ψ: R[x] → C defined by

ψ(p(x)) = p(1 + 3i).

In Exercises 9.42 to 9.47, are the polynomials irreducible in the given ring? Give reasons.

9.42.x3 + x2 + x + 1 in Q[x].

9.43.3x8 − 4x6 + 8x5 − 10x + 6 in Q[x].

9.48.Is Z[x] a euclidean ring when δ(f (x)) = degf (x) for any nonzero polynomial? Is Z[x] a euclidean ring with any other definition of δ(f (x))?

9.49. Can you define a division algorithm in R[x, y]? How would you divide x3 + 3xy + y + 4 by xy + y3 + 2?

9.50. Let Lp be the set of all linear functions f : Zp → Zp of the form f (x) = ax + b, where a = 0 in Zp . Show that (Lp , Ž ) is a group of order p(p − 1) under composition.

9.51. If p is a prime, prove that (x − a)|(xp−1 − 1) in Zp [x] for all nonzero a

9.55.Is 5 irreducible in Z[i]?

9.56.Show that Z[√−5] = {a + b√−5|a, b Z} does not have the unique factorization property.

9.57.Prove that a gaussian integer is irreducible if and only if it is an invertible element times one of the following gaussian integers:

(1)any prime p in Z with p ≡ 3 mod 4.

(2)1 + i.

(3)a + bi, where a is positive and even, and a2 + b2 = p, for some prime p in Z such that p ≡ 1 mod 4.

9.58.If r/s is a rational root, in its lowest terms, of a polynomial p(x) with integral coefficients, show that p(x) = (sx − r)g(x) for some polynomial g(x) with integral coefficients.

9.59.Prove that r/s, in its lowest terms, cannot be a root of the integral polynomial p(x) unless (s − r)|p(1). This can be used to shorten the list of possible rational roots of an integral polynomial.

9.60.Let m = m1m2 · · · mr and Mi = m/mi . If gcd(mi , mj ) = 1 for i = j , each of the congruences Mi y ≡ 1 mod mi has a solution y ≡ bi mod mi . Prove that the solution to the simultaneous congruences

x ≡ a1 mod m1, x ≡ a2 mod m2, . . . , x ≡ ar mod mr

For Exercises 9.61 to 9.64, solve the simultaneous congruences.

164−216 1862 469

9.66.Solve the following simultaneous equations:

26x − 141y = −697

55x − 112y = 202

(a) in Z2, (b) in Z3, and (c) in Z5. Then use the Chinese remainder theorem to solve them in Z assuming they have a pair of integral solutions between

Find its value without using a calculator. (If you get tired of doing arithmetic, calculate its value mod 10 and mod 11 and then use the Chinese remainder theorem.)

9.68.The polynomial x3 + 5x Z6[x] has six roots. Does this contradict Theorem 9.6?

9.69.If R is an integral domain and R[x] is euclidean, show that R must be a field.

9.70.Assume that R is a euclidean domain in which δ(a + b) max{δ(a), δ(b)} whenever a, b, and a + b are all nonzero. Show that the quotient and remainder in the division algorithm are uniquely determined.

For example, (n) = nZ, consisting of all integer multiples of n, is the principal ideal generated by n in Z.

The set of all polynomials in Q[x] that contain x2 − 2 as a factor is the principal ideal (x2 − 2) = {(x2 − 2) · p(x)|p(x) Q[x]} generated by x2 − 2 in Q[x]. The set of all real polynomials that have zero constant term is the principal ideal (x) = {x · p(x)|p(x) R[x]} generated by x in R[x]. It is also the set of real polynomials with 0 as a root.

The set of all real polynomials, in two variables x and y, that have a zero constant term is an ideal of R[x, y]. However, this ideal is not principal (see Exercise 10.30).

However, every ideal is principal in many commutative rings; these are called principal ideal rings.

Theorem 10.2. A euclidean ring is a principal ideal ring.

Proof. Let I be any ideal of the euclidean ring R. If I = {0}, then I = (0), the principal ideal generated by 0. Otherwise, I contains nonzero elements. Let b be a nonzero element of I for which δ(b) is minimal. If a is any other element in I , then, by the division algorithm, there exist q, r R such that

I (b).

Conversely, any element of (b) is of the form q · b for some q R, so q · b I . Therefore, I (b), which proves that I = (b). Hence R is a principal ideal

Proposition 10.4. Let I be ideal of the ring R. If I contains the identity 1, then I is the entire ring R.

Proof. Let 1 I and r R. Then r = r · 1 I , so I = R.

Let I be any ideal in a ring R. Then (I, +) is a normal subgroup of (R, +), and we denote the coset of I in R that contains r by I + r. Hence

I + r = {i + r R|i I }.

The cosets of I in R are the equivalence classes under the congruence relation modulo I . We have

r1 ≡ r2 modI if and only if r1 − r2 I.

By Theorem 4.18, the set of cosets R/I = {I + r|r R} is an abelian group under the operation defined by

(I + r1) + (I + r2) = I + (r1 + r2).

In fact, we get a ring structure in R/I .

Theorem 10.5. Let I be an ideal in the ring R. Then the set of cosets forms a ring (R/I, +, ·) under the operations defined by

(I + r1) + (I + r2) = I + (r1 + r2)

and

(I + r1)(I + r2) = I + (r1r2).

This ring (R/I, +, ·) is called the quotient ring (or factor ring) of R by I .

Proof. As mentioned above, (R/I, +) is an abelian group; thus we only have to verify the axioms related to multiplication.

R, then

(I + r1){(I + r2)(I + r3)} = (I + r1)(I + r2r3) = I + r1(r2r3) = I + (r1r2)r3

= (I + r1r2)(I + r3) = {(I + r1)(I + r2)}(I + r3).

Also,

(I + r1){(I + r2) + (I + r3)} = (I + r1){I + (r2 + r3)} = I + r1(r2 + r3)

=I + (r1r2 + r1r3) = (I + r1r2) + (I + r1r3)

={(I + r1)(I + r2)} + {(I + r1)(I + r3)}.

The other distributive law can be proved similarly. The multiplicative identity is I + 1. Hence (R/I, +, ·) is a ring.