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