課程名稱︰代數導論二
課程性質︰系必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012年3月22日
考試時限（分鐘）：15:30~18:20 (170分鐘)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Algebra 2 Exam 1
03/22 2012
Write your answer on the answer sheet. We give partial points.
In this examination. R denotes a ring, F denotes a field, R[X](resp. R[X,Y])
denotes polynomial ringsover R in variable X(resp. X,Y). Let F_p denotes the
finite field of order p.
(1)(42 points) "yes" or "no". Either give a brief reason or give a counter
example. 7 points each.
(a) The ring Z[X] is a unique factorization ring.
(b) Every ideal in Q[X,Y] is principle.
(c) The polynomial 1+X+......+X^22 is irreducible in Q[X].
(d) The element √2+√3 is the root of a degree a polynomial in Q[X].
(e) The element √2+√3 is the root of a degree a polynomial in Q(√6)[X].
(f) If α€ Q(√2), then either Q(α)=Q or Q(α)=Q(√2).
(2)(40 points) Prove the following assertions. 10 points each.
(a) For each prime number p, there exists an irreducible quadratic polynom-
ial in F_p[X].
(b) The ring Q[X]/(x^3+22) is actually a field.
(c) The ring F_7[X]/(x^11-4) consists of exactly 7^11 elements.
(d) If f(X)€ F[X] is of degree d, then in F there exist at most d roots of
f(X).
(3)(10 points)
Let A be a finitely generated abelian group written additively. For each
prime number p, let pA := {px | x€A}. Show that there exists a non-negative
integer d such that for almost all p, the quotient group A/pA is isomorphic
to A1⊕...⊕Ad, where each Ai is a cyclic group of order p (Hint: write A
as athe direct sum of cyclic groups).
(4)(8 points)
Let M denote the direct sum of the Q[X]-module M1 = Q[X]/(x-1) and
M2 = Q[X]/(x+1). Can M be generated by just one element? Prove or disprove
it.
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