课程名称︰代数导论二 课程性质︰必修 课程教师︰陈其诚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2012/04/26 考试时限（分钟）：195 mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Write your answer on the answer sheet. We give partial points. In this examination, Fp denote 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 regular pentagon is constructible (by straightedge/ruler and conpass). (b) The regular 9-gon is constructible (by straightedge/ruler and conpass). (c) The regular 10-gon is constructible (by straightedge/ruler and compass). (d) The number 2cos(2π/7) is a root of x^3 + x^2 -2x -1. (e) Let ω be a root of x^2 + x + 1. Then Q(ω) is the splitting field of x^4 + x^2 + 1 over Q. 3 (f) The field Q(√5) has no automorphisms other than the identity automorphism/map. 3 注：√5 = 5^(1/3) (2) (40 points) Prove the following assertions. 10 points each. (a) If p is a prime number, then the splitting field over Q of the polynomial x^p - 1 is of degree p - 1. (b) Let E be an extension of a field F and let f(x) ∈ F[x]. If ψ is an automorphism of E leaving every element of F fixed, then ψ takes a root of f(x) in E. (c) There exists an irreducible polynomial of degree 2 over Fp. (d) There exists an automorphism ψ of Q(√3,√5) such that ψ(√3) = -√3 and ψ(√5) = -√5. (3) (18 points) Let f(x) = x^3 + ax + b ∈ Q[x] be irreducible over Q and let K be its splitting field over Q so thar α,β,γ ∈ K are the three roots of f(x). (a) (9 points) Let δ := (α-β)(β-γ)(γ-α). Show that δ^2 ∈ Q. (b) (9 points) Show that the degree of K/Q is 3 if and only if δ ∈ Q. --

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