【站僕】摩檸Morning>試卷(2016/09/08)

# 98 年 - 98 淡江大學 轉學考 代數#55938

【非選題】
1.1, (a) (10 poinlis) Let a，b  Z and d - gcd{a, fc}. Show that there exists r s  Z that ra+sb=d.

【非選題】
2.(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.

【非選題】
3.(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.

【非選題】
4.
2. (15 points) Let

【題組】(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,

【非選題】
5.【題組】(b)
such that

【非選題】
6.【題組】(c)

【非選題】
7..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.

【非選題】
8.(h) Construct a finite field of 27 eloinctits.

【非選題】
9.
4. (20 poinl.s) Let G =- <a> be a cyclic grovip of order n.

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

【非選題】
10.【題組】(b) Show filial; if (r, n) = d、tlion ad  <ar>

【非選題】
11.【題組】(c) Show that, tbn subgroup <ar> has order

【非選題】
12.5. (15 points) (a) Explain why non-trivial group homomorphismexisf

【非選題】
13.(b) Show (Jiaf; a nori- trivial group bomomorpbism exhibitrig an example.

【非選題】
14.
6 (15 points) Let R be an integral clomam. .

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

【非選題】
15.【題組】(b) Suppose ihnl R is a PID. Show tlmt, every irreducible element in /i! is a prime.

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