98年 - 98 淡江大學轉學考代數#55938

科目：轉學考-代數 | 年份：98年 | 選擇題數：0 | 申論題數：15

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所屬科目：轉學考-代數

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申論題 (15)

【已刪除】1, (a) (10 poinlis) Let a，b

Z and d - gcd{a, fc}. Show that there exists r s

Z that ra+sb=d.

(b) (5 points) Let a，b and c bo, intege.vs. If a and c arc relatively prime, show that, c | ab itnplies that- c | b.

【已刪除】(c) (5 points) Show that, p

Z is a prime if and only for all integers a and b, p | ah implies p | a or p 丨 b.

(a) Show that if a² + b² is a prime in % iJicn a b√-l is a prime in R. Give an example f,o show I,hah the convcrae is not. trvio,

【已刪除】(b)

【已刪除】(c)

.3. (15 points) (a) Construct a field F over Q such l-hal; x⁷ +2x + 2 has a root, in F. Find the degree of extension of F over Q.

(h) Construct a finite field of 27 eloinctits.

(a) Show f,hat for any d | n, there is a subgroup of order d,

【已刪除】(b) Show filial; if (r, n) = d、tlion a^d

<a^r>

【已刪除】(c) Show that, tbn subgroup <a^r> has order

【已刪除】5. (15 points) (a) Explain why non-trivial group homomorphism

exisf

【已刪除】(b) Show (Jiaf; a nori- trivial group bomomorpbism

exhibitrig an example.

(a) Show llmt every prime olcrrient in R is irrexlnciblo,

(b) Suppose ihnl R is a PID. Show tlmt, every irreducible element in /i! is a prime.