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）︰2019/11/8 考試時限（分鐘）：180 試題 : (滿分100分) (以下的屬於符號都用ε代替) 1. (10%) Show that 〈(1,2),(1234)〉=S4. Is it true that〈(1,2),(1324)〉=S4? 2. (10%) Let p1, p2,..., p2019 be distinct primes. Prove that any group of order p1p2...p2019 can be generated by 2019 elements. (Hint: Use Lagrange's theorem.) 3. (10%) Show that lim φ(n)=∞ , where φ is the Euler's phi function. n->∞ 4. (20%) Let G be a group and H, K ne its finite subgroups. We define a relation ~ on the Cartesian product H×K by the rule:(h,k)~(h',k') if and only if hk=h'k' in G. (i) Show that the relation ~ is an equivalence relation. (ii) Given (h,k)εH×K, determine [(h,k)], the equivalence class containing (h,k), and the order of [(h,k)]. |H||K| (iii) Show that |HK|=--------- , where HK:={hk|hεH, kεK}. |H∩K| (iv) Is HK necessarily a subgroup of G? Prove it or give a counterexample. 5. (10%) Let p be a prime and F be a field with |F|=p^3. We know that F has Zp as a subfield. Given aεF, show that there exists a cubic polynomial f(x)εZp[x] such that f(a)=0. 6. (20%) (i) (10%) Take any matrix AεM2(C) that is not a scalar multiple of the identity matrix. Prove that BεM2(C) commutes with A if and only if B is of the form B=αA+βI for some scalars α,βεC. (ii) (10%) Let S be a complex subalgebra of M2(C) such that for all a,bεS we have ab=ba. What is the largest possible dimension of S? 7. (10%) Find the smallest positive integer n such that n≡1 (mod 7) n≡7 (mod 8) n≡5 (mod 9). 8. (10%) Let 0 1 0 M=( 0 0 1)εM3(R). 0 0 0 (i) Determine the subring of M3(R) generated by M. (ii) Determine the R-subalgebra of M3(R) generated by M. --

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