課程名稱︰代數導論二
課程性質︰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）︰2014/05/15
考試時限（分鐘）：30分鐘
試題 :
代數導論第五次小考
INTRODUCTION TO ALGEBRA II - QUIZ V
MAY 15 2014
(1) (15 points) Please determine the splitting field and its degree over Q
for x^p - 2, where p is prime.
(2) (15 points) Prove that [f(x)]^p = f(x^p) for any polynomial f(x) ∈ F_p[x].
(3) (20 points) First recall the theorem [Dummit-Foote,Section 13.4,Exercise5]:
Let K be a finite extension of F. Then K is splitting field over F for some
g(x) ∈ F[x] if and only if every polynimal in F[x] with a root in K splits
completely in K[x].
Now let K_1 and K_2 be finite extensions of F contained in the field K.
Assume they are both splitting field over F. Prove that K_1∩K_2 is a
splitting field over F.