102年 - 102 淡江大學轉學考代數#53069

科目：轉學考-代數 | 年份：102年 | 選擇題數：0 | 申論題數：9

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所屬科目：轉學考-代數

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申論題 (9)

【已刪除】 (a) Suppose p is a prime number and a is an integer, (a,p) — 1. Prove that a^p~1

1 (mod p).

(b) What is the remainder when 35³⁵ is divided by 37?

2. (12 pts) Prove or disprove: If G is a group of order 53, then G must be cyclic.

3. (12 pts) Suppose G = {e, a, 6, c} is a group of order 4; but it contains no element of order 4. Write out the operation table for G.

(a) Prove that every finite integral domain is a field.

(b) Give an example of an integral domain which is not a field.

5. (12 pts) Show that the principal ideal (x — 1) in Z[x] is prime but not maximal. .

(a) Show that x³ + x + 1 is irreducible in Z₅ [x].

(b) Let R be the quotient ring Z₅[x]/ (x³ +x + 1). How many elements are there in R1 Is R a field? Please justify your answer