MODERN ALGEBRA (MATH 210) MIDTERM

All rings are commutative rings with 1. The problems are not in order of increasing difficulty. (In particular, 4(b) is in some sense a “bonus” problem; you may want to leave it until you’re done with the others.) The problems have equal value. If you have any questions, just ask.

1. Show that a ring in which all ideals are finitely generated cannot have an infinite sequence of ideals

I_{1} ⊊ I_{2} ⊊ I_{3} ⊊ ….

Conversely, show that if a ring has no infinite sequence of ideals, then all ideals are finitely generated.

2.

(a) Show that if n ≠ 4, the only normal subgroups of S_{n} are {e}, A_{n}, and S_{n}.

(b) Describe all group homomorphisms from S_{7} → S_{5}.

3. Prove that every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.

4.

(a) Note that ω = \frac{−1+\sqrt{−3}}{2} is a cube root of 1, and ω^{2} + ω + 1 = 0. Prove that the subset {x + yω ∈ ℤ[ω] : x + y is divisible by 3 } ⊂ ℤ[ω] is an ideal. Is it prime?

(b) Describe the set of integers of the form a^{2} − ab + b^{2} (a, b integers). (Hint: Note that N(a + bω) = a^{2} − ab + b^{2}.)

5. Find (with proof) an ideal I of ℤ[i] whose quotient F is a field of 9 elements. Is there an ideal whose quotient is a field of 25 elements?

6. How many abelian groups are there of order 288?

Date: Thursday, November 7, 2002.