課程名稱︰代數導論二
課程性質︰數學系大二必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰103/04/17
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (15 points) Prove that Z[i] is a Principal Ideal Domain. (You can assume
the fact that a Euclidean domain is a PID without proof.)
(2) (15 points) Let R be a ring with identity. An element e ε R is called
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an idempotent if e = e. Assume e is an idempotent in R and er = re for all
r ε R. Prove that Re and R(1-e) are two-sided ideals of R and R is
isomorphic to Re × R(1-e). (You may assume Chinese Remainder Theorem.)
(3) (15 points) Let R be a Unique Factorization Domain (UFD) with quotient
field F and let p(x) ε R[x]. Prove that if p(x) = A(x)B(x) for some
nonconstant polynomials A(x), B(x) ε F[x], then there exists r,s ε F
such that rA(x) = a(x) and sB(x) = b(x) both are in R[x] and p(x) =
a(x)b(x) is a factorization in R[x].
(4) (15 points) Assuming Eisenstein's criterion, show that
6 5 4 3 2
(i) x + x + x + x + x +8x + 1 is irreducible in Q[x], and
n
(ii) X - x is irreducible in (Q[x])(X).
(5) (10 points) Let ψ(n) be the Euler ψ-function. Prove that Σψ(d) = n,
d|n
where the sum is over all the divisors d of n.
3
(6) (10 points) Show that x + x + 1 is irreducible over F_2. (You don't have to
prove the irreducibility criterion you are applying, but you need to state
the criterion explicitly in order to obtain full point.)
Let θ be a root. Please compute θ^n in F_2(θ) for all n ε Z ≧ 0.
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(7) (15 points) Show that √2 is not in any finite field entension K over Q of
degree k, which is not divisible by 3.
(8) (15 points) Let F be a field of characteristic not equal to 2. Show that any extenstion of F of degree 2 is of the form F(√D), where D is an
element of F which is not a square in F.
Remark: There are 110 points totally.
註：Z代表整數環；ε代表屬於符號；Q代表有理數環；F_2代表order為2的體