課程名稱︰代數導論二
課程性質︰必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2012/04/26
考試時限（分鐘）：195 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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
註：√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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