189-235A: Basic Algebra I Practice Final Exam

Fall 2004

This mock exam has twelve questions, worth 9 points each. The final grade will be taken out of 100, although it is possible to achieve a maximum grade of 108.

1. Let (u_{n})_{n≥0} be the Lucas sequence, defined recursively by the rule

u_{0} = 0, u_{1} = 1, u_{n+1} = 2u_{n} + u_{n−1}, for n ≥ 1.

Show that u_{n} ≤ 3^{n} for all n ≥ 0.

2. Solve the congruence equation

4x = 11 (mod 55).

3. Let a be an element of a group G, and suppose that a^{m} = a^{n} = 1, for some integers m and n. Let d = gcd(m, n). Show that a^{d} = 1. (You may use any of the properties of the gcd that were shown in class.)

4. Let Z_{3} denote the field with 3 elements. Show that the greatest common divisor of the polynomials x^{3} −1 and x^{2} +1 in Z_{3}[x] is equal to 1, and express 1 as a linear combination of these two polynomials.

5. Show that the ring Z_{3}[x]/(x^{2} + 1) is a field. Write down the multiplicative inverse of the element [x^{3} − 1] in this field.

6. Give an example of three rings R_{1}, R_{2} and R_{3} of cardinality 25 which are not isomorphic to each other. (You should justify your assertion.)

7. Show that the ring Q[x]/((x−1)^{2}) is not isomorphic to the ring Q[x]/((x^{2}− 1).

8. Construct an injective group homomorphism from GL_{2}(Z_{3}) to S_{8} (the symmetric group on 8 elements.)

9. Give an example of a ring R and an ideal I which is prime but not maximal. Give a maximal ideal J of R which contains I.

10. Write down two non-isomorphic groups of order 10.

11. Let F be a field, and let G be the set of functions f : F ⟶ F of the form f(x) = ax + b, with a ∈ F − {0} and b ∈ F. Show that G is a group under the operation of composition of functions. Let H be the subgroup of translations, i.e. functions of the form f(x) = x + b for some b ∈ F. Show that H is a normal subgroup of G and that G/H is isomorphic to the multiplicative group F^{×} of non-zero elements of F.

12. Let F be a field. State (without proof) whether the following assertions are true or false. a. The set I of polynomials p(x) in the ring F [x] satisfying p(1) = p(2) = 0 is an ideal of F [x]. b. The set I of polynomials of degree at most 3 is an ideal of F [x]. c. Every ideal in Q[x] is principal. d. Every ideal in Z is principal. e. The symmetric group S_{n} has cardinality n. f. The group S_{5} contains an element of order 6. g. The group S_{6} contains an element of order 7.