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題組內容

4.

(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.

相關申論題

(a) [5%] n= 2021 = 43 x 47.

(b) [15%] n = 2020 = 4 x 5 x 101.

(i) A has only one maximal ideal.

(ii) AI m consists of units in A.

(iii) If a,b are not unit, then a+ b is not a unit.

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

(b) [10%] Determine the cardinality of invertible 2 x 2 matrices with coefficients in Z/nz in terms of Φ(n).

(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).

(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

(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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