课程名称︰代数导论 课程性质︰必修 课程教师：陈荣凯 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰980615 考试时限（分钟）：150分钟 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : (1)(20 pts) Let k be a field and R=k[x,y] be the ploynomial ring with two indeterminates. Let p=(x) be the ideal generated by x. (a) Show that p is a prime ideal. (b) Show that p is not a maximal ideal. (c)Let Rp:={f/g | f(x,y),g(x,y) in R[x,y] ,g isn't in p=(x)} be the localization and let m = {x*h/g | h(x,y),g(x,y) in R,g isn't in p=(x)}. show that Rp/m is automorphic to K(y). (Hint: Consider the homomorphism Rp → K(y) by sending f(x,y)/g(x,y) to f(0,y)/g(0,y).) (2)(15 pts) Let K/f be an extension of degree p, whrer p is a prime number . Show that for all u in K-F , we have K=F[u} and deg(irr(u,F))=p. (3)(25 pts) Determine the Galois group of X^3-5 over Q and all the intermediate subfields. (4)(15 pts) Let K be a finite field of P^n elements, where p is a prime number. We regard K as an extension over Zp. (a)Show that the polynomial x^(p^n)-x is a separable polynomial in Zp[x]. (b)Show that every element in K satisfies x^(p^n)-x in Zp[x]. (c)Show that K is Galois over Zp. (5)(25 pts) Let ξ= e^(2πi/5) be the 5-th root of unity. We consider the extention Q[ξ] over Q. (a)(10pts) Show that irr(ξ,Q)=x^4+x^3+x^2+x+1. (b)(5 pts) Express (2ξ+1)^(-1) in terms of a3ξ^3+a2ξ^2+a1ξ+a0 for some ai in Q. (c)(10 pts) Solve ξby radicals. (Hint: consider the intermediate field Q[u], where u = ξ+ξ^(-1).) --

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