Final Exam¶

Question

(1) 定理叙述 (20 pt)
(i) 叙述 Sylow 第一定理和 Sylow 第二定理.
(ii) 叙述环的对应定理.

(2) 作业题 (40 pt)
(i) If \(\lvert G \rvert = p^nq\), with \(p > q\) primes, then \(G\) contains a unique normal subgroup of index \(q\).
(ii) (a) If \([F : K]\) is prime, then there are no intermediate fields between \(F\) and \(K\).
(b) lf \(u \in F\) has degree \(n\) over \(K\), then \(n\) divides \([F : K]\).
(iii) (a) If \(F\) is a field then every nonzero element of \(F[[x]]\) is of the form \(x^ku\) with \(u \in F[[x]]\) a unit.
(b) \(F[[x]]\) is a principal ideal domain whose only ideals are \(0, F[[x]] = (1_F) = (x^0)\) and \((x^k)\) for each \(k > 1.\)
(iv) Let \(A\) be a cyclic \(R\)-module of order \(r \in R\).
(a) If \(s \in R\) is relatively prime to \(r\), then \(sA = A\) and \(A[s] = 0\).
(b) If \(s\) divides \(r\), say \(sk = r\), then \(sA \cong R/(k)\) and \(A[s] \cong R/(s)\).

(3) (10 pt) 给出 \(392\) 阶交换群的完全分类 .

(4) (10 pt) 设 \(f = x^4 + x^3 + x^2 + x + 1\). \(\alpha\) 是 \(x\) 在 \(\mathbf{Z}[x]/(f)\) 中的剩余类 . 将 \((\alpha^3 + \alpha^2 + \alpha)(\alpha^5 + 1)\) 用 \(\{1, \alpha, \alpha^2, \alpha^3\}\) 线性表示 .

(5) (10 pt) 称非零 \(R-\) 模 \(M\) 为单模 , 若 \(M\) 只有两个子模 \(0\) 和 \(M\).
(i) 证明: 任意 \(R-\) 单模均同构于 \(R/M\), 其中 \(M\) 是 \(R\) 的极大理想.
(ii) 证明 Schur 引理: 设 \(\phi: S \to S'\) 为 单模的模同态, 则 \(\phi\) 为零同态或同构.

(6) (10 pt) 设 \(R\) 是实系数三角多项式环 \(\mathbf{R}[\sin x, \cos x]\). 证明 :
(i) \(R\) 中元素 \(f\) 都有如下的形式:

其中 \(a_0, a_m, b_m \in \mathbf{R}\).
(ii) 设 \(\operatorname{deg}_{tr} f\) 为使得 \(a_r, b_s \neq 0\) 的最大值, 则

(iii) \(R\) 中没有零因子 .
(iv) \(\sin x, 1 - \cos x\) 是 \(R\) 的不可约元.
(v) \(R\) 不是唯一分解整环.