課程名稱︰代數導論二 課程性質︰數學系必修 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）︰106/4/21 考試時限（分鐘）：120分鐘 試題 : 1.(15 points) Recall that in a PID R, the ideal I = (r)⊂R generated by r ∈R is maximal if and only if r is irreducible in R. Show that C[x,y] is not a PID. ~ × 2.(15 points) Let R be an integral domain and R = R ∪{0} the collection ~ of units of R together with 0. An element u ∈R - R is called a universal ~ side divisor if for every x ∈R there is some z ∈R such that we have x = qu + z for some q ∈R. Show that if R is a Euclidean domain that is not a field, then there exists a universal side divisor in R. 3 3.(15 points) Let θbe a root of the polynomial x -3x-3. 1 + θ 2 Please express -------------- as a Q-linear combination of {1 ,θ ,θ} 2 3 + 2θ+ θ in Q(θ). 4.(15 points) Let K be a extension of the field F. Prove that K is a finite extension over F if and only if K is generated by a finite number of algebraic elements over F. 5.(15 points) Let R be a UFD with quotient field F and let p(x)∈R[x]. Prove that if p(x)=A(x)B(x) for some nonconstant polynomials A(x),B(x)∈F[x], then there exist r,s ∈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]. 6.(15 points) Show that 10 9 8 7 6 5 4 3 2 (i)x + x + x + x + x + x + x + x + x + 12x + 1 is irreducible in Q[x],and n (ii)X - t is irreducible in (C(t))[X], where n is an integer larger than 1. (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 points.) 3 7.(15 points) Show that x - √2 is irreducible in Q(√2)[x]. 8.(15 points) Show that Q(√2,√3) = Q(√2+√3). --

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