| 91年 6月19日 10:30~12:00 應用數學系 (選考) | 誠實是我們珍視的美德， 我們喜愛「拒絕作弊，堅守正直」的你！ |
| 科目： 代 數 |
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1.
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Prove that a finite semigroup has a subsemigroup which is a group. (10%) | |
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2.
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Prove that a nonabelian group has at least six elements. (10%) | |
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3.
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Let H and K be isomorphic normal subgroups of a group G. Prove or disprove : G/H and G/K are isomorphic (10%) | |
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4.
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Construct a field with 243 elements. (10%) | |
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5.
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Let F be a finite field. Prove that the multiplicative group 〈F\ {0}, •〉 is a cyclic group. (10%) | |
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6.
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(a) | Give the definitions of UFD (unique factorization domain) and PID (principal ideal domain). |
| (b) | Is every UFD a PID ? State your reason. (10%) | |
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7.
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Prove that every group of order 255 is cyclic. (10%) | |
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8.
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Let R be a commutative ring with unity. Prove that | |
| (a) | Every maximal ideal is a prime ideal. | |
| (b) | Is every prime ideal a maximal ideal ? | |
| State your reason. (20%) | ||