Groups, Rings and Fields: Problem sheet 5 Solutions must be submitted by Tuesday 14 January 2003

1. In each case, find the all the units in the ring Z_{n}. For each unit find its inverse and order in the group U(Z_{n}): (i) n = 12, (ii) n = 15, (iii) n = 16, (iv) n = 20, (v) n = 24. [15 marks]

2. In each of the following commutative rings R, determine whether the element a is a unit. If not determine whether it is a zero divisor:

(i) R = Z[\sqrt{7}] = \{b + c\sqrt{7} : b, c ∈ Z\}, a = 8 + 3\sqrt{7}; (ii) R = Z[\sqrt{7}], a = 3 + \sqrt{7}; (iii) R = C(R), the ring of continuous functions on R, and where a is given by a(x) = 1 + x^{2}; (iv) R = C(R), a(x) = x − |x|. [11 marks]

3. Show that the ring Z[\sqrt{7}] has infinitely many units. [6 marks]

4. Consider the ring Z[\sqrt{−2}] = \{a + b\sqrt{−2} : a, b ∈ Z\} (where \sqrt{−2} is shorthand for the complex number i\sqrt{2}). Show that v defined by v(α) = |α|^{2} is a Euclidean function on Z[\sqrt{−2}]. [10 marks]

5. Consider the ring R = Z[\sqrt{−6}] = \{a + b\sqrt{−6} : a, b ∈ Z\}. Define N(a + b\sqrt{−6}) = |a + b\sqrt{−6}|^{2} = a^{2} + 6b^{2}. Observe that N(αβ) = N(α)N(β). By considering the function N, show that 2, 3, and ±\sqrt{−6} are all irreducible in R. From the equation

6 = 2 × 3 = (\sqrt{−6})(−\sqrt{−6})

deduce that R does not have unique factorization. [8 marks]

6. In each case use the Euclidean algorithm to find g = gcd(a, b). Also find r, s ∈ Z with g = ra + sb:

(i) a = 37, b = 21; (ii) a = 102, b = 75; (iii) a = 375, b = 213; (iv) a = 893, b = 511;

(v) a = 1001, b = 777. [15 marks]

7. (i) Use the Euclidean algorithm in Q[X] to find a greatest common divisor of the polynomials f(X) = X^{5} − 2X^{3} − X − 1 and g(X) = X^{4} − X^{3} − 3X^{2} + 2X + 2.

(To simplify the computations you may use MAPLE. The MAPLE command rem calculates the remainder left by dividing a polynomial by another polynomial. See the online help for syntax and examples of use. Although MAPLE can actually compute the answer directly, for credit you must show the intermediate steps.) (ii) Use the Euclidean algorithm in Z_{2}[X] to find a greatest common divisor of the polynomials f(X) = X^{6} + 1 and g(X) = X^{5} + X^{4} + 1. (iii) Use the Euclidean algorithm in Z_{3}[X] to find a greatest common divisor of the polynomials f(X) = X^{4} + 2X^{3} + 2X^{2} + X + 1 and g(X) = X^{4} + X^{3} + X + 2. [20 marks]

8. In each case, use the Euclidean algorithm in Z[i] to find a greatest common divisor γ of α and β. Also find λ, μ ∈ Z[i] with γ = λα + μβ.

(i) α = 20 + 9i, β = −2 + 3i; (ii) α = 44 + 32i, β = 23 + 13i; (iii) α = 193, β = 81 + i. [15 marks]

RJC 2/12/2002