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 2 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. 3 (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 --

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