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100年 - 100 淡江大學 轉學考 代數#55909
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3. Let G be a group such that | G | ＜ 200. Suppose G has subgroups of order 33 and 55, find the order of G..
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(a) 2549 is divided by 11
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2. Show that a group G is abelian if and only if x2= e for any x in G, where e is the identity of G.
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(c) [8%] Let f(x) = xn-x -1 for n≥ 2. Show that if a monicsatisfies
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(b) [8%] Show that for n ≥ 2 and f(x) = xn -x - 1, the two polynomials have no root in common in C.
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(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
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(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).
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(b) [10%] Determine the cardinality of invertible 2 x 2 matrices with coefficients in Z/nz in terms of Φ(n).
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(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.
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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
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