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中国科学技术大学 w/b>awI
2016年秋季博士资格考试试卷 D A<F{n.Z:
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代数学 A6Ghj{~
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1." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1.1.(40分) 考虑形式幂级数环 C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]={a0+a1x+a2x2+⋯∣ai∈C}C[[x]]={a0+a1x+a2x2+⋯∣ai∈C} 考虑 2" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">22 阶全矩阵环 R=M2(C[[x]])" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">R=M2(C[[x]])R=M2(C[[x]])． si_W:mLF{a
(1) 证明 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 为 Noether 整环； c_
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(2) 描述 C[[x]]" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">C[[x]]C[[x]] 全部的有限生成不可分解模，并给出论证； " -w ^D!C
(3) 给出环 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 全部的双边理想，并给出论证； t#C,VwMe[
(4) 描述 R" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">RR 上全部的有限生成不可分解左模，以及这些模的自同态环． YE zU{J
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2." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">2.2.(40分) 将 Abel 群与 Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ZZ-模等同起来，考虑 Abel 群 G=Z3⊕Z" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=Z3⊕ZG=Z3⊕Z． otO
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(1) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部子群，并给出论证； i+g~ Uj}h
(2) 将 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的每个商群都分解成不可分解群的直和，并给出论证； 93j{.0]X
(3) 列出群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的全部直和项，并给出论证； bW53" `X
(4) 描述 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的自同构群． "!O 1j
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回顾：Abel 群 G" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">GG 的子群 A" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">AA 称为直和项，若存在另一子群 B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">BB 满足 G=A+B" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">G=A+BG=A+B 以及 A∩B={0}" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">A∩B={0}A∩B={0}． r'/\HWNP
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3." role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">3.3.(20分) 具体给出代数同构 6B0# 4Qrv
CS3→∼C×C×M2(C)，" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; text-align: left; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3−→～C×C×M2(C)，CS3→～C×C×M2(C)， 其中 CS3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">CS3CS3 为 S3" role="presentation" style="word-wrap: normal; outline: none; display: inline; line-height: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">S3S3 的群代数；并给出相应的论证． (89Ji'dc
提示：利用不可约复表示． ?~J i-{#X
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