課程名稱︰代數導論二
課程性質︰數學系大二必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰103/06/16
考試時限（分鐘）：150
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (15 points) Please state the first 3 parts (G1-G3) of the Fundamental
Theorem of Galois Theory. (In the rest of the test G1-G5 could all be
assumed.)
(2) (15 points) Prove that Q(√2+√3) = Q(√2, √3). Show that [Q(√2+√3) : Q]
= 4 and find the minimal polynomial of √2 + √3 over Q.
4
(3) (15 points) Determine the splitting field and its degree over Q for x + 2.
(4) (15 points) Let p(x) be an irreducible polynomial over a field F of
positive characteristic. Then there exists an integer k ≧ 0 and an
irreducible separable polynomial p_sep(x) ε F[x] such that
k
p
p(x) = p_sep(x).
(5) (15 points) Consider the polynomial f(x) = x - 4x -1 ε Z[x]. Show that
f(x) is irreducible by reducing the polynomial to F_3[x]. Please compute
the Galois group of f(x) over Q and then determine its solvability by
radicals.
(6) (15 points) Let K be a degree n cyclic field extension over F, which
n
contains n-th roots of unity. Prove that K = F(√a) for some a ε F.
(7) (15 points) Let K/F be a Galois extension and F'/F be any extension. Prove
that KF'/F' is Galois. Define the map π:Gal(KF'/F') → Gal(K/F) by
π(σ) = σ|_K . Show that π is a well-defined injective group
homomorphism.
(8) (15 points) Let ζ = exp(2πi/17) be the primitive 17-th rooth of unity.
2 8 9 15
Show that ζ + ζ + ζ + ζ lies in a degree 4 field extension over Q.
Remark: There are 120 points totally.
註：Q代表有理數體；ε代表屬於符號；Z代表整數環；F_3代表order為3的體；σ|_K代表σ限制於K上