臺灣大學數學系

九十一學年度第一學期碩博士班資格考試題

 代數 (Algebra)

Sept 14, 2002

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• (1)(15$\%$) Let $A$ be a real tridiagonal matrix (that is, an $n \times n$ matrix all of whose entries are zero except possibly those of the form $a_{i-1,i},a_{ii},a_{i,i-1}$. Show that if $a_{i,i-1}$ and $a_{i-1,i}$ are both positive, both negative, or both zero for $i=2,\cdots, n$, then $A$ has real eigenvalues. (Hint: A diagonal matrix $D$ can be found such that $DAD^{-1}$ is real symmetric.)

  
  

• (2)(15$\%$) Let $V$ be a finite dimensional vector space and $T : V \rightarrow V$ be a linear transformation. Let $W$ be an invariant subspace (that is, $TW \subset W$). Let $m(t),m_1(t)$ and $m_2(t)$ be the minimal polynomial of $T$ as linear transformation of $V,W,V/W$, respectively. Show that $m_i(t) \vert m(t),i=1,2$, and $m(t) \vert m_1(t)m_2(t)$.

  
  

• (3)(20$\%$) Let $N,H,K$ be normal subgroups of a group $G$. (a) Suppose $NK=HK$ and $N \cap K=H \cap K$. Is $N=H$ necessarily true? (b) Does $G$ necessarily contain a subgroup isomorphic to $G/N$ ? (c) Suppose $G/N \cong G/H$. Is $N \cong H$ necessarily true? (d) Suppose $G/N \cong G/H$ and $N \subseteq H$. Is $N=H$ necessarily true?

  
  

• (4)(15$\%$) Let $p$ be a prime. Show that for any $a \in Z /p Z$, there exist $b,c \in Z /p Z$ such that $a=b^2+c^2$.

  
  

• (5)(15$\%$) Describe all subrings (containing 1) of $Q$.

  
  

• (6)(20$\%$) Let $K_0= Q$. If $i \ge 0, K_{i+1}$ is the smallest subfield of $C$ containing the set $\{ a \in C \vert a^n \in K_i for some n>0 \}$. Let $K=\bigcup _{i=0}^{\infty}K_i$. (a) Prove that $K$ is a field. (b) Let $f(x) \in K[x]$ be irreducible. Prove that deg $f(x) \ne 2,3,4$.

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