(i) Show that aKa−1 ⊂ H

FINAL FOR ALGEBRA

Date: 2000, January 17, 9:10–11:00AM Each of the following problems worth 10 points.

1.

(i) Give an example of a group of order 4 which is not cyclic.

(ii) Give an example of a infinite non-abelian group.

(iii) Give an example of a non-abelian solvable group.

(iv) Give an example of a non-commutative division ring.

(v) Give an example of an ideal I of a commutative ring R such that I is prime but not maximal.

2.

(i) What is the characteristic of the ring Z6× Z? why?

(ii) What is the commutator subgroup of a simple non-abelian group? Why?

(iii) What is the order of the element (12)(345)(12) in S₈? Why?

3. Suppose that H is a normal subgroup of a group G and K is a normal subgroup of H. Let a be an element in G.

(i) Show that aKa⁻¹ ⊂ H.

(ii) Show that aKa⁻¹ is a normal subgroup of H.

4.

(i) Find all prime number p such that x + 2 is a factor of x⁴+ x³+ x²− x + 1 in Zp[x].

(ii) Show that for p a prime, the polynomial x^p+ a in Zp[x] is not irreducible for any a ∈ Zp. 5. Show that φ : C → M2(R) given by

φ(a + bi) =

"

a b

−b a

#

for a, b ∈ R gives an isomorphism of C with the subring φ[C] of M2(R) where M2(R) is the ring of two by two matrices over R.

6.

(i) Is Q[x]/hx²− 5x + 6i a field? Why?

(i) Is Q[x]/hx²− 6x + 6i a field? Why?

7. Let A and B be ideals of a ring R. The product AB of A and B is defined by AB =n X_n

i=1aibi

¯¯

¯ ai∈ A, bi∈ B, n ∈ Z⁺ o

. (i) Show that AB is an ideal of R.

(ii) Show that AB ⊆ (A ∩ B).

8. Let R be a commutative ring and N be an ideal of R. Define

√N = { a | aⁿ∈ N for some n ∈ Z⁺}.

(i) Show that N ⊆√

N and√

N is an ideal of R.

(ii) Give an example of N such that√

N = N .

1

2 FINAL FOR ALGEBRA

(ii) Give an example of N such that√

N 6= N . 9.

(i) Let K be a subgroup of index 2 of a group G. Suppose that a ∈ G − K and b ∈ G − K i.e., a, b are in G but not in K. Show that ab ∈ K.

(ii) Let G be a finite abelian group. Suppose that G has two distinct elements of order 2. Show that 4 divides |G|.

10. Let φ : R → R be a nontrivial ring homomorphism.

(i) Show that φ(a) = a if a ∈ Z.

(ii) Show that φ(a) = a if a ∈ Q.

(iii) Show that φ[R⁺] ⊆ R⁺ where R⁺= { a ∈ R | a > 0 }. (Hint: a square is positive.) (iv) Show that φ(a) > φ(b) if a, b ∈ R and a > b.

(v) Show that φ(a) = a for all a ∈ R.