課程名稱︰代數導論一
課程性質︰系定必修
課程教師︰黃漢水 教授
開課系所︰數學系
考試時間︰2007/01/18 12:20-15:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Let Z be the set of all integers, Q be the set of all rational numbers
and R be the set of all real numbers.
一 Let F be an ordered fiedld. (20%)
(1) If a in F and a > 0 then a^(-1) > 0 .
(2) If a in F and 1>a>0 then a^(-1) > 1 .
(3) If a in F and 0>a>-1 then a^(-1) < -1 .
二 Let f(x) = x^3 + 3x^2 + 3x + 2 , g(x) = x^3 + 4x^2 + 2x + 1 be polynomials
in Z_5[x] . (15%)
(1) Find the g.c.d. of f(x) and g(x) .
(2) Find the l.c.m. of f(x) and g(x) .
a
三 Let D = { ── │ a,n in Z, n≧0} and Z（ D（ Q . (25%)
45^n
(1) Find the set U = {c│c in D, c is a unit in D} .
(2) If I is an ideal in D and J = I∩Z ,
then prove or disprove that J is an ideal in Z .
(3) Find all prime ideals I of D. What is D／I ?
(4) Find the g.c.d. of 4620 and 1260 in D .
四 Let D = { ┌a c┐ │ a,b,c,d in Q} and I be an ideal of D . (20%)
└b d┘
(1) If a,b,c d in Q and ┌a c┐ in I, then prove or disprove that
└b d┘
┌a 0┐ in I, ┌b 0┐ in I, ┌c 0┐ in I, ┌d 0┐ in I .
└0 0┘ └0 0┘ └0 0┘ └0 0┘
(2) If I≠0 then prove or disprove that I = D .
五 A rectangular prism 3 cm long with 1 cm square ends is to have each of
its eight corners tipped with one of n colors. (20%)
(1) How many distinguishable prism are possible if each color may be used
on any number of corners ?
(2) How many distinguishable prism are possible if no color is to be
repeated on different corners ?
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