Dept. Math, NCKU Qualifying Exam: Algebra Oct 2017

Question: 1 2 3 4 5 6 7 Total

Points: 10 10 15 20 15 15 15 100

Score:

Easier: 1,2 Medium: 3,4,5 Harder: 6,7

1. (10 points) Prove that any group of order 45 is not simple.

2. (10 points) Show that the ideal generated by 7 and x^{3}− 2 in Z[x] is maximal.

3. (a) (10 points) Find all intermediate fields between Q and Q(√ 2,√

3).

(b) (5 points) Show that √

5 is not an element Q(√ 2,√

3).

4. (a) (10 points) Prove that every PID is a UFD.

(b) (10 points) Show that Z[√

−1] is a UFD.

5. (15 points) Let F be a finite field of order p^{n}where p is a prime and n a positive integer. Prove
that there is exactly one subfield of order p^{m} for each divisor m of n.

6. (15 points) Let M be a finite generated module over a commutative ring R with identity and φ : M → M be an R-module homomorphism. Prove that there exists a polynomial p(x) in R[x]

such that p(φ) = 0 as an element in the R-algebra Hom_{R}(M, M ) of R-module homomorphisms.

7. (15 points) Let p be a prime and let G be a finite group whose Sylow p-subgroup is normal.

Show that the number of elements of order p in G is congruent to −1 modulo p.