課程名稱︰代數導論一 課程性質︰數學系大二必修 課程教師︰李秋坤教授 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2019/12/20 考試時限（分鐘）：50 試題 : (滿分100分) (以下的屬於符號都用ε代替) 1. (a) (10%) Determine Z(D8), the center of the Dihedral group D8. (b) (10%) Find all subgroups of D8. 2. (20%) Show that, for each hεH, dimR R[h]≦2. (Here H is the Quaternions, the 4-dimensional real algebra.) 3. (a) (10%) Let G be a cyclic group of order n and let dεN. Show that if d divides n then G contains a subgroup of order d. (b) (10%) Show that A4 has no subgroup of order 6. (Hint: Suppose that H is a subgroup in A4 of order 6. Show that there exists a 3-cycle σ not belongs to H. And then consider H, σH, and (σ^2)H.) 4. (10%) Suppose that m and n be relatively prime positive integers. Determine whether there exists a non-trivial group homomorphism from the additive group Zm to the additive group Zn or not, and verify your answer. 5. Let R and S be rings with identity elements 1R and 1S, respectively. Suppose that φ: R→S is a nonzero additive map such that φ(xy)=φ(x)φ(y) for all x,yεR. (a) (10%) Show that if φ is surjective then φ(1R)=1S. (b) (10%) Show that if there is an element uεR such that φ(u) is invertible in S then φ(1R)=1S. (c) (10%) Show that if S has no zero-divisors then φ(1R)=1S. --

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