課程名稱︰代數二 課程性質︰數學系選修，可抵必修代數導論二 課程教師︰林惠雯 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017.4.17 考試時限（分鐘）：180分鐘 試題 : ALGEBRA(II)— MIDTERM APRIL 17, 2017 1.(40%) Let [L：K]＜∞. Show that the following statements are equivalent. (a) L is a Galois extension of K; (b) L is a splitting field of an irreducible separable polynomial f(x) over K; (c) L is a splitting field of a separable polynomial f(x) over K; (d) |Aut(L/K)| = [L：K]; Aut(L/K) (e) K = L ; (f) There is a one-to-one correspondence between subfields of L/K and subgroups of Aut(L/K); G (g) K = L for some finite subgroup G of Aut(L). 2.(25%) 7 (a) Determine the Galois group G of the splitting field L of x －1 over Q and the correspondence between subgroups of G and subextensions of L/Q. (Justify your answer.) (b) Let K = Q(ζ) with ζ a primitive 6-th root of unity. Set 2 3 f(x) = (x －2)(x －2) and let L be the splitting field of f over K. Find a such that L = K(a); determine [L：K], and show that L is a cyclic extension of K. (Justify your answer.) p p 3.(12%) Let x and y be indeterminates and let L = F (x,y), K = F (x ,y ) p p with p a prime. Show that L does not have a primitive element over K. 4.(25%) Let L = Q(cosπ/9) and K = Q. (a) Show that L is a splitting field of a separable polynomial f(x) and find Gal(L/K). (Justify your answer.) (b) Show that L/K is not an extension by radicals. (c) Find a Galois extension by radicals E/K such that L ⊂ E. (d) Find the Galois group Gal(E/K) of your example. 5.(18%) Give ONE polynomial f(x) ∈ Q[x] of degree ≧ 4 such that its Galois group over Q is isomorphic to S . (Justify your answer.) deg f --

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