臺灣大學數學系

八十八學年度第一學期碩博士班資格考試試題

代數

[回上頁]

1. Show that every group of order 175 is abelian.
2. Show that the additive groups of $\mathbb{R}$ and $\mathbb{C}$ are isomorphic.
3. If $R$ is a ring (with 1), then the ideals $I$ and $J$ are called coprime if $I+J=R$ . If $I_i$ (i=1,2,3,4,5) are coprime in pairs, show that $I_1I_2$ and $I_3I_4I_5$ are also coprime.
4. If $f:A\rightarrow A$ is an $R$-module homomorphism such that $f\circ f=f$, show that
   $A$ = Ker $f \oplus$ Im$f$.
5. Let $\sigma_1,\sigma_2,\cdots ,\sigma_n$ be distinct nonzero homomorphisms from a field $K$ into a field $L$. Show that the ${\sigma_i}'s$ are linearly independent over L, i.e. that if $a_1,a_2,\cdots ,a_n\in L$ and $a_1\sigma_1(x)+a_2\sigma_2(x)+ \cdots +a_n\sigma_n(x)=0$ for all $x\in K$, then $a_1=a_2=\cdots =a_n=0$.
6. Show that there is no automorphism of $R$ other than the identity mapping.
7. Let $V$ be a vector space of dimension 20. If $V_1,V_2,V_3,V_4$ are subspaces of dimension 9, 12, 10, 13 respectively, and $W=(V_1\cap V_2)+(V_3\cap V_4)$,
   1. find max{dim $W$} and the conditions for the max to be attained.
   2. same for min{dim$W$}.
8. Find the rational canonical form of the matrix
   
   $$\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 4 & 0 & 0 & 0 \\ \end{array}\right]$$
   　

   (i) over $\mathbb{Q}$, (ii) over $\mathbb{R}$, (iii) over $\mathbb{Z}/(17)$.

[回上頁]