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 x3− 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 pnwhere p is a prime and n a positive integer. Prove that there is exactly one subfield of order pm 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 HomR(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.
![A partition P of [a, b] is a finite collection of points of [a, b] such that P contains a, b](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)