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）︰2019/10/18 考試時限（分鐘）：50 試題 : (滿分100分) (以下的屬於符號皆用ε代替) 1. (20%) Find all finite nonempty subsets of Z that are closed under multiplication. 2. (20%) Suppose a finite Abelian group G does NOT contain elements x≠1 such that x^2=1. Determine whether |G| is even or odd. 3. (20%) Determine the subring of R×R generated by the element (0,π). 4. Let N0:=N∪{0} be the set of all nonnegative integers. (a) (4%) Given any (a,b), (c,d)εN0×N0, we say (a,b)~ (c,d) if a+d=b+c. Show that ~ is an equivalence relation on N0×N0. (b) (4%) Let S:= {[(a,b)]|(a,b)εN0xN0} be the set of all equivalence classes in N0×N0 with respect to ~. Define two binary operations ☆ and ★ on S by [(a,b)]☆[(c,d)]:=[(a+c,b+d)] and [(a,b)★(c,d)] :=[(ac+bd, ad+bc)] for [(a,b)], [(c,d)]εS. Show that ☆ and ★ are well-defined maps. (c) (4%) Show that S is an Abelian group under ☆. (d) (4%) Show that S is a monoid under ★. (e) (4%) Show that S is a ring under the addition ☆ and the multiplication ★. 5. (a) (7%) Show that any subfield of C contains Q. (b) (7%) Determine the subfield F of C generated by √2 and i. (c) (6%) Let F as in (b). Then, by (a), F can be viewed as a vector space over Q. Compute dimension of F over Q and find its basis. --

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