Final Exam
Math 6121A (Algebra I)
Dec 13, 2000.
Time: 2hrs 50 min.

Attempt all questions. You are advised to spend not more than one hour on Part A.

Part A

Unless explicitly directed you do not have to justify your answers when answering questions in this part.

1.

How many groups are there of orders:

(a)

4:

(b)

6:

(c)

15:

(d)

19:

(e)

49:

2.

Give an example of a group having a proper subgroup isomorphic to itself.

3.

Are any two Jordan-Holder series for a group G necessarily identical ?

4.

If the (multi-set of) factor groups appearing in Jordan Holder series of two groups are identical then the groups are isomorphic. True or false ?

5.

The (multi-set of) factor groups of two Jordan-Holder series of the same group must be the same. True or false ?

6.

List the orders of all elements of the dihedral group D₈ ?

7.

State the number of conjugacy classes (orbits under the action of conjugation) of the following groups:

(a)

The cyclic group of order n.

(b)

The dihedral group D₆ (of order 6).

8.

What are the possible numbers 3-Sylow subgroups in a group of order 72 ?

9.

Give a presentation of the following groups by generators and relations:

(a)

The dihedral group D_2n.

(b)

The cyclic group Z_n.

10.

Give an example of a group G, having a subgroup H and an element $g \in G$, such that $gHg^{-1} \subset H$, but $g Hg^{-1} \neq H .$

11.

What are the Betti numbers and torsion coefficients for the finitely generated Abelian group presented by the following matrix ?

$$\left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right)$$

12.

Describe the fiber product in the category of sets.

13.

In the category of A-modules, state which of the following functors are co-variant and which are contra-variant.

(a)

${\mbox Hom}_A(M,\cdot)$:

(b)

${\mbox Hom}_A(\cdot, N)$:

(c)

$M \otimes_A \cdot$:

14.

Let A be the ring of $2 \times 2$ real matrices, $V = {\Bbb R}^2$, with elements of V written as column vectors. Then, A acts on V by the usual matrix multiplication, making V an A-module.

(a)

Is V a free A-module ?

(b)

Is V a projective A-module ?

Give one sentence explanations for each of your answers above.

15.

Let $0 \rightarrow N' \rightarrow N \rightarrow N'' \rightarrow 0$ be an exact sequence of A-modules. Let M be another A-module. What conclusions can you draw about exactness after applying the following functors:

(a)

${\mbox Hom}_A(M,\cdot)$,

(b)

${\mbox Hom}_A(\cdot,M)$,

(c)

${\mbox Hom}_A(M,\cdot)$, with the additional information that M is a projective module.

16.

For each of the following rings state if they are factorial, and also if they are principal.

(a)

k[x,y] where k is any field.

(b)

k[x]

(c)

${\Bbb Z}[x,y]$

(d)

${\Bbb Z}[x]$

17.

Give an example of a ring that is principal but not an integral domain.

18.

Let V be an n-dimensional k-vector space. What is the dimension of the exterior algebra $\oplus_i \wedge^i(V)$ as a k-vector space ?

19.

The tensor product of two free modules is always a free module. True or false ?

20.

Let U,V be two finite dimensional k-vector spaces of dimension m,n respectively. What is the dimension of $U \otimes_k V$ as a k-vector space ?

21.

Give simpler descriptions (i.e. without involving tensor products) of the following tensor products:

(a)

$k[X] \otimes_k k[Y]$,

(b)

${\Bbb Z}/9{\Bbb Z} \otimes_{\Bbb Z} {\Bbb Z}/16{\Bbb Z}$, where ${\Bbb Z}/9{\Bbb Z},{\Bbb Z}/16{\Bbb Z}$ are considered as ${\Bbb Z}$-modules.

22.

Let R be a Noetherian ring.

(a)

Is R[X] a local ring ? Is it Noetherian ?

(b)

Is R[[X]] a local ring ? Is it Noetherian ?

23.

Does a quotient of a Noetherian ring have to be Noetherian ?

24.

Does a subring of a Noetherian ring have to be Noetherian ?

25.

Which amongst the following are fields:

(a)

${\Bbb Q}[X]$,

(b)

${\Bbb Q}(X)$,

(c)

${\Bbb Q}[\sqrt{2}]$.

26.

What is the cardinality of the algebraic closure of the finite field with two elements ?

27.

Is the field of complex numbers the algebraic closure of ${\Bbb Q}$ ?

28.

Is the field of complex numbers the algebraic closure of ${\Bbb R}$ ?

29.

What is the degree of the extension of the finite field F_qⁿ over F_q ?

30.

What is the size of the automorphism group of the finite field of order pⁿ, where p is a prime ?

31.

Give an example of a separable extension that is not a normal extension.

Part B

1.

Prove that there exists no simple group of order 30.

2.

Let G be a finite group and, for each prime p, choose a p-Sylow subgroup of G. Prove that G is generated by these subgroups (that is every element of G is expressible as a product of some elements of these subgroups.)

3.

Let A be a commutative ring and I an ideal distinct from A. Let $J = \{ x \in A \vert x^n \in I, \;\;\mbox{for some}\; n > 0\}.$Prove that J is an ideal and is equal to the intersection of the prime ideals of A containing I.

4.

Find a matrix that presents the ideal generated by (x²,y³)as an A-module.

5.

Let f be a polynomial of degree n with coefficients in a field k. Let L be a splitting field of f over k. Prove that [L:k] is a divisor of n!.