課程名稱︰代數導論一
課程性質︰必修
課程教師︰陳其誠
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/12/28
考試時限（分鐘）：120
試題 :
Write your answer on the answer sheet. You should include in your
answer every piece of reasonings so that correspinding partial credit
could be gained.
Part I. True or False. Either prove the assertion or disprove it by a
counter-example (6 point each):
(1) There is no x∈Z satisfying simultaneously x ≡ 2 (mod 10)
and x ≡ 8 (mod 12).
(2) The congruence equation 4^x + 307^y ≡ 1 (mod 1228) has no
integer solution for x, y.
Hint: 1228 = 4 x 307 and 307 is a prime number.
(3) There is no degree 2 irreducible polynomial in F_5[x].
(4) The polynomial x^2+x+1 is the only degree 2 ireeducible
polynomial in F_2[x].
(5) The polynomial xy-zw is ireeducible in Q[x,y,z,w].
Part II. For each of the following problems, give accordingly a short
proof or an example (8 points each):
(1) The kernel of the ring homomorphism Z[x] → C defined by
x |-> √2 is a principal ideal.
(2) The ring Z[√-2] is a unique factorization domain.
(3) The ring Z[√-5] is not a unique factorization domain.
(4) x^4+x+1 is irreducible in F_2[x].
(5) A UFD is not necessary a PID. ━
(6) If α∈Z[i] has no integer factor and αα is a square integer,
then α is a square in Z[i].
Part III. Give a complete proof or a complete calculation (10 points each):
(1) Factorize 3x^5+6x^4+9x^3+3x^2-1 in Q[x].
(2) Find the greatest common divisor of 3+4i and 4+7i.
(3) Show that Z[i]/(5) is isomorphic to F_5 x F_5.