[試題] 105-2 林惠雯 代數二 第一次小考
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）︰2017/3/20
考試時限（分鐘）：50分鐘
試題 :
1. Do one of the following problems.
(a) Let f(x) ∈ K[x]. Prove that a splitting field of f(x) over K exists and
is unique up to isomorphism.
(b) Let K be a field. Prove that an algebraic closure of K exists and is
unique up to isomorphism.
(c) Let K ⊂ M ⊂ L be a tower of fields. Prove that L/K is a separable
extension if and only if both L/M and M/K are separable extensions.
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2. Determine the Galois group G of the polynomial x ＋2 over Q and the
correspondence between subgroups of G and intermediate fields between Q and
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the splitting field L of x ＋2 over Q.
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