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110年 - 110 國立臺灣大學_碩士班招生考試_數學研究所:代數#102176
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題組內容
2. [20%] Let A be a commutative ring with identity, and m a maximal ideal. Show the following statements are equivalent:
(ii) AI m consists of units in A.
相關申論題
(a) [5%] n= 2021 = 43 x 47.
#429737
(b) [15%] n = 2020 = 4 x 5 x 101.
#429738
(i) A has only one maximal ideal.
#429739
(iii) If a,b are not unit, then a+ b is not a unit.
#429741
3. [20%] Prove the following simple form of the structure theorem for finitely generated mod- ules over a principal ideal domain (so you ca can not apply the struc cture theo orem directly). Let A be a principal ideal domain and M a 2 x 2 matrix whose entries are in A. Show that there exist invertible matrices P, Q with entries in A and a,β A with a | β such that
#429742
(a) [5%] For a positive integer n, let Φ(n) denote the cardinality of invertible elements in the ring Show that where p runs through all primes dividing n.
#429743
(b) [10%] Determine the cardinality of invertible 2 x 2 matrices with coefficients in Z/nz in terms of Φ(n).
#429744
(c) [5%] Determine the cardinality of invertible 2 x 2 matrices with coeficients in Z/n2 whose determinants are equal to 1 in terms of Φ(n).
#429745
(a) [4%] Let be a monic with f(0) = ±1 such thathave no common root in C. Suppose f(x)=g(x)h(x)for non n-constant.Show that there exists a monic
#429746
(b) [8%] Show that for n ≥ 2 and f(x) = xn -x - 1, the two polynomials have no root in common in C.
#429747
相關試卷
110年 - 110 國立臺灣大學_碩士班招生考試_數學研究所:代數#102176
110年 · #102176
102年 - 102 淡江大學 轉學考 代數#53069
102年 · #53069
100年 - 100 淡江大學 轉學考 代數#55909
100年 · #55909
98年 - 98 淡江大學 轉學考 代數#55938
98年 · #55938
96年 - 96 淡江大學 轉學考 代數#55974
96年 · #55974
94年 - 94 淡江大學 轉學考 代數#56429
94年 · #56429
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