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國立高雄師範大學九十 六學年度碩士班招生考試試 題
系所別：數學系
科
目 ： 代 數 （ 包 括 線 性 代 數 及 代 數 學 ）（ 全 一 頁 ）
一 、 Let u1  (1,0,2,1), u2  (1,1,2,0), u3  (2,0,5,4), v1  (1,0,1,0), v2  (0,1, 1,0), v3  (1,1,0,0),
and u  (1,1,3, 2) . Suppose that T : R 4  R 4 is a linear operator and that
T (ui )  vi i  1,2,3.
(a) (8％) Find T ( u ) .
(b) (8％) Is T one-to-one? Explain your answer.
(c) (8％) Let U be the orthogonal complement of span{v1, v2 , v3} . Find an
orthogonal basis of U .
二 、 Determine each of the following statements is true or false. If true, prove it;
if false, give a counterexample.
(a) (8％) If A is an n  n real matrix then there is a real number  such
that A   I is invertible.
(b) (8％) Let A be an n  n real matrix. If A2 is diagonalizable then A
is diagonalizable.
(c) (10％) Let A be an m  n matrix and B be an n  m matrix. If m  n
and AB  I m then BA  I n .
Do all problems and show your reasoning. Answers without explanation may receive no credit.
三、Let G be a group and Z(G) ={a  G | ga=ag for all g  G} be the center of G.
(a) (5％) Let N be a normal subgroup of G of order 2. Show that N  Z(G).
(b) (5％) Show that if the quotient group G/Z(G) is cyclic, then G is abelian.
(c) (5％) Let G be the dihedral group of order 8 and write G=
a,b| a4=b2=1, ba=a-1b
. Find
Z(G) and point out which well-known group is isomorphic to the quotient group G/Z(G) .
四、(10％) Prove that no group of order 992 is simple. [992=25*31.]
五、(10％) Prove that in a commutative ring with identity, every nonzero prime ideal of finite index is maximal.
六、Let Q be the field of rational numbers and let K be a splitting field of (x2-3)(x3-2) over Q. Note that
K=Q( 3 2 , 3 ,  ), where  =(-1+i 3 )/2 is a primitive 3rd root of unity in C.
(a) (5％) Show in detail that the degree [K: Q] is 12.
(b) (5％) Give generators for the Galois group G of K over Q.
(c) (5％) Find an element   K such that Q ( 3 2 , 3 ) = Q (  ).