Let S¹ = {z ∈ C* | |z| = 1} and Un = {z ∈ C* | z" = 1} for n ∈ N. Check that U_n ⊆ S¹. Further, show that U_n ≤ S¹.

Let F be a field and let f(x) ∈ F[x] be irreducible in F[x]. Show that the ideal (f(x)) is a maximal ideal in F[x]. Use this to deduce that Q[x]/(x² + 6x3 +12) is a field.

Find the order of each of the elements in U(15). Is U (15) cyclic? Justify your answer.

If H and K are normal abelian subgroups of a group, and if H∩ K = {e}, show that HK is abelian. Will the result be still true if we remove the condition that H and K are normal? Justify your answer.

Find the gcd of the polynomials x4 + 3x + 2 and x³ + 3x² + 5x + 3.

Let   . Check whether Check whether R is a subring of M₂(R). Is R an ideal of M₂(R)? Justify your answer.

Show that, if G is a finite group and a ∈ G, o(a) | o(G). Further, show that a°(G) = e for all a ∈ G. Deduce the Euler-Fermat theorem a(n) = 1 (mod n) for all a, n ∈ N, n ≥ 2, (a, n) = 1.

Calculate the following:

(i) (3x² + 4x + 1) + (3x3 + 4x² + 2x + 3) in Z₅[x].

(ii) (3x²+2x+6). (3x3+4x+5) in Z₇[x].

Define an integral domain. Give an example of an integral domain which is not a field.

Let R be a ring in which a² = a for all a ∈ R. Show that a = -a and R is commutative.

Let R = Z₂₀

(i) Give, with justificaiton, a nilpotent element in R.

(ii) Give, with justification, a zero divisor in R which is not nilpotent.

(iii) What is the order of U(R)?

If F is a field, show that U(F[x]) = F*.

Let a = (125), β = (1432) ∈ Ss. Compute σ = α β-¹. Write as a product of transpositons. What is the signature of σ?

Let A be a 3 x 4 real matrix, B be a 4 × 2 real matrix and C be a 2 x 3 real matrix. Which of the following operations are defined?

(i) CA + B¹

(ii) AB + C¹

For those operations that are defined, what is the order of the resulting matrix?

Let S = {1, 2, 3, 4} and ∗ be the binary operation defined by a ∗ b = a. Compute the Cayley table for (S, ∗). Is ∗ commutative? Is ∗ associative? Justify your answers.

State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?

Define a semigroup. Give an example of an infinite semigroup.

Define a subgroup of a group. Check whether 

is a subgroup of the the group of 2 × 3 matrices over ℂ under addition

Define an abelian group. Give an example of a non-abelian group. (You don’t need to prove that your example is a group. You have to only prove that it is non-abelian.)

Discuss the role of evaluation in RRR by highlighting its various types.