中原大學九十學年度博士班入學招生考試

6月13日　　第2節 　　　數學系	誠實是我們珍視的美德， 我們喜愛「拒絕作弊，堅守正直」的你！
科目：代數	（共1頁 第1頁）

Instructions. Work precisely five (5) of the ten problems. Do at least one problem on groups (problems 1-4); Do at least one problem on rings (problems 5-7); do at least on problem on fields (problems 8-10).

	Read over the examination before you start to write.
1.	(20) Let G be a group of order p2q where p≠q for positive primes p and q. Show that G is not simple (that is, that G has a subgroup H subject to 1＜ H＜ G.)
2.	(20) Let H be a normal subgroup of a finite group G whose order and index in G are relatively prime : (∣H∣, [G:H]) = 1. Show that H is the only subgroup of G of order ∣H∣.
3.	(20) Let G be any group, finite or infinite. Define G' to be the subgroup of G generated by all elements of the form x-1y-1xy, x G, y G. Prove that： (a) G' is normal in G. (b) G / G' is abelian. (c) If H is a normal subgroup of G such that G/H is abelian, then G' H .
4.	(20) Let A be an abelian group, and let P be the set of elements of A of finite order. (a) Prove that P is a subgroup of A and that all non-identity elements of A/P has infinite order. (b) Show by example that the set of all elements of A of infinite order, do not necessarily form a subgroup of A.
5.	(20) Define a maximal ideal M of a ring R to be an ideal properly contained in R such that if I is any ideal with the property M I R, then I = R. (a) Prove that if R has a unit element, then it has a maximal ideal. (b) Give an example of a ring with no maximal ideal,
6.	(20) Let R be a commutative ring with a unity. (a) Prove that if R is a principal ideal domain (P.I.D.) then R is a unique factorization domain (U.F.D.). (b) Show that the converse of Part(a) of this problem is false. (c) Give an example of a commutative integral domain with a unity which is not a U.F.D.
7.	(20) Let R be a ring with no non-zero nilpotent right ideals. Suppose that N is a non-zero minimal ideal of R. Prove that N = eR for some idempotent e ( e≠0, e2 = e).
8.	(20) (a) Show that a field K has pn elements (where p is a prime) if and only if K is a splitting field over Zp , the field GF(p) of p elements, of the polynomial -x. (b) Explain how the result of part (a) can be used to prove the existence and uniqueness of the field with pn elements.
9.	(20) Let M/K be a normal field extension. Let f K[x] be a monic irreducible polynomial over K. Suppose that f has a root in M. Show that f factors over M into a product of distinct linear factors.
10.	(20) Let F be a finite field, F* the non-zero elements of F. Prove that F* is a cyclic group under the field multiplication.

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