[試題] 108-1 李秋坤 代數導論一 第一次小考
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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