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︰2006/11/13 10:20-11:20
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
一 Let G be a group. If H and K are normal subgroups of G and
H∩K = {e}
Prove that for any a屬於H, b屬於K, ab = ba . (30%)
二 Let R be a ring such that for any a屬於R, a^2=a .
(1)Prove that for any a屬於R, a+a = 0 . (10%)
(2)prove that R is a commutative ring . (20%)
三 Let Q be the field of all rational numbers and
m(x) = x^2 - 3 屬於 Q[x]
(1)If f(x)=ax+b屬於Q[x] and f(x)≠0,
then find a polynomial g(x)=cx+d屬於Q[x] such that
(ax+b)(cx+d)≡1(mod m(x)) . (25%)
(2)Find a polynomial h(x)=px+q屬於Q[x] such that
(x+2)(px+q)≡1(mod m(x)) . (15%)
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