NAME:.................................. 18.701 Final exam.

This is a closed book exam. Do not forget to put your name on this exam. Please write clearly. Explain your answers (except if you are asked not to explain).

1. Define a ring. Define a homomorphism of rings.

2. Let R be a ring.

(a) Define a nilpotent of R.

(b) Assume R is a domain and x ∈ R. Define what is a factorization of x.

3. Let G be a finite group. Define a normal subgroup of G.

4. True or false: The following module M is isomorphic to a free module over the ring R. (No explanation necessary.)

(a) R = ℤ[X] and M = ℤ[X]/(5X + 1).

(b) R = ℤ[i] and M = {(x, y) ∈ ℤ[i] × ℤ[i] : (5i + 1)x + (i − 1)y = 0}.

(c) R = ℤ and M is the module defined by generators x, y, z and relations 3x+4y−z = 0, 11x + z = 0.

5. Suppose that M is a module over R = 𝔽_{7}[X] which has a presentation with the following presentation matrix:

( \begin{matrix}0 & 1 & X^{2} \\ −X & 0 & X^{2} − X \\ X & X & 0 \end{matrix} ) .

(a) Give the structure of M as in the structure theorem for modules over a Euclidean ring.

(b) Think of M as a vector space over 𝔽_{7}. What is the dimension of M as an 𝔽_{7} vector space?

(c) Think of multiplication by X as a linear operator on M seen as an 𝔽_{7} vector space. Does there exist a basis for M such that the matrix of this operator is diagonal?

6. Let I be the kernel of the ring homomorphism φ : ℚ[X, Y] → ℂ which maps X to \sqrt{17} − 1 and Y to \sqrt{5/2}.

(a) What is the image of the map φ?

(b) Find generators for the ideal I.

(c) Is I a maximal ideal?

7. Groups and automorphisms.

(a) Let g be an element of the group G. Let λ_{g} be the inner automorphism of G given by conjugation with g. Show that the order of λ_{g} (as an element of the automorphism group of G) divides the order of g.

(b) Give an example of a g and G where the order of λ_{g} is not equal to the order of g.

8. Prove GL_{2}(𝔽_{29}) has an element of order 8 using the following steps.

(a) Factor X^{4} + 1 into two monic irreducible polynomials over 𝔽_{29}. Call them f and g.

(b) Why is F = 𝔽_{29}[X]/(f) a field?

(c) Find a basis of F = 𝔽_{29}[X]/(f) as a vector space over 𝔽_{29}.

(d) Show that X̄ is an element of the multiplicative group F^{∗} of F which has order 8. (Hint: X̄^{4} + 1 = 0.)

(e) Find the matrix of multiplication by the element X̄ with respect to the basis you found in (c).

(f) Conclude that GL_{2}(𝔽_{29}) has an element of order 8.

NAME:.................................. 18.701 Final exam.

9. Find a 2-Sylow subgroup and a 3-Sylow subgroup of the permutation group S_{4}.

10. Prove that the set of elements in S_{n} with a fixed cycle structure forms a conjugacy class. [Recall that (123)(56) has the same cycle structure as (14)(235) but not the same as (12345).]