推 acherub:最後一題老師題目寫錯，Fermat prime : p = 2^(2^m)+1 07/01 10:32
課程名稱︰代數導論二
課程性質︰大二必修
課程教師︰李白飛
開課系所︰數學學系
考試時間：2006/6/26 10:20-12:50
是否需發放獎勵金：是
試題 :
Algebra II
1. Determine the Galois group and all the subfields of the splitting field of
x^4+2x^2+4 over Q. (15pts)
2. Show that a group of order p^2q is solvable where p,q are distinct primes.
(15pts)
3. Show that, for any odd n, the alternating group A_n is generated by
n-cycles. (15pts)
4. Determine whether the equation 2x^5-4x^4+1=0 is solvable over Q.
5. Show that a division ring, as a vector space over its center, cannot be
3-dimensional. (10pts)
6. (a) Show that 2 is a primitive root of unity modulo 29.
(b) Solve x^6+x^5+x^4+x^3+x^2+x+1≡0 (mod 29). (10pts)
7. Given a prime 1847, determine whether x^2≡365 (mod 1847) is solvable.
(10pts)
8. Let p>3 be a Fermat prime, that is, p=2^m+1 for some m>1. Show that 3 is
a primitive root of unity modulo p.(15pts)
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 61.217.159.69