私立中原大學八十八學年度博士班招生考試命題紙

系組別：數學系博士班

科目：代數

考試時間：

06月09日第2節

代數學考題

1. (20%)Let　Aut(G)　denote the set of all automorphisms of a group G.　Prove that

   (a)	If G is an infinite cyclic group,　then　Aut(G)　is a group od order 2.
   (b)	If G is a cyclic group of order n,　find　|Aut(G) |.

   
   
2. (10%)If G is a nonabelian group of order 6,　then .
   　
3. (10%)Let F be a field and .　Show that forms a maximal ideal of F[x].
   　
4. (10%)Let R be a principal ideal domain.　Is the polynomial ring R[x] also a principal ideal domain? why?
   　
5. (10%)Let R be a commutative ring with identity.　Show that ,Hom_R(R,R) and R are ring isomorphisms,　where
   Hom_R(R,R) denotes left R-module homorphisms from R into itself.
   　
6. (10%)If F is a finite field, then F\{0} is a cyclic group under multiplication.
   　
7. (10%)Let E be the splitting field in the complex numbers C of the polynomial f(x)=x³-2 over the rational numbers Q.
   Characterize the Galois group Aut_QE.
   　
8. (20%)Let Z[i] be Gaussian integers,　.
   Show that

   (a)	Z[i] is a Euclidean domain.
   (b)	Find a greatest commom divisor of　7 + i　and　3 － 4i.

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