課程名稱︰代數導論一
課程性質︰系定必修
課程教師︰莊正良 教授
開課系所︰數學系
考試時間︰ (不知)
試題： ┌────────────────┐
│Ｚ: the ring of integers │
│Ｒ: the ring of reals │
│Ｃ: the ring of complex numbers │
└────────────────┘
1.(10%) Find all integers x, y between 1 and 200 satisfying 149x - 68y = 1.
2.(15%) Determine whether the following two pairs of rings are isomorphic
or not. Give your reason.
(a) Ｚ4 x Ｚ6 and Ｚ24. (5%) (b) Ｃ and Ｒ[x]/(x^2 + 1) (10%).
(註：(a)中數字為下標，Ｚ4 x Ｚ6 的"x"是"乘")
3.(15%) Let F = Ｚ2[x]/(x^2 + x + 1). (註：Ｚ2[x] 的 2 為下標)
(a) Show that F is a field by writing down its multiplcation table. (5%)
(b) Factorize t^4 - t in F[t]. (10%)
4.(15%) Consider the subring R = {a + b√2: a, b ∈Ｚ} of Ｃ.
(a) Show that f: a + b√2 → a - b√2 is a ring isomorphism of R.
(b) Show that the only isomorphisms of R are f and the identity map.
(c) Find all units of R. (5% each)
5.(15%) Find all intergers n > 0 such that any nonzero element of Ｚn is
either a unit or a nilpotent element. (註：Ｚn 的 n 為下標，第7.題同理)
6.(15%) Let m, n > 0 be relatively prime intergers. Let a, b be integers
such that a≠b (mod mn). Show that the equation (x-a)(x-b) = 0 has at least
four distint solutions in Ｚmn. Hint: Use Chinese Remainder Theorem.
(註：Ｚmn 的 mn 為下標)
7.(15%) An element e of a ring is called an idempotent if e^2 = e. Find
integers n≧0 such that the only idempotents of Ｚn are 0 and 1.
Hint: You may use the result of problem 6.
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