台灣大學數學系

九十一學年度第二學期博士班資格考試題

代數

 

May 10, 2003

 

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1.  

By a rotation of the 3-dimensional real space $R^3$, we mean a rotation about a line through the origin. Let $A$ be a linear transformation of $R^3$. Suppose that $A$ preserves the Euclidean distance in the sense that $\vert\vert Ax\vert\vert=\vert\vert x\vert\vert$ for all $x\in$ $bR^3$. (1) If the determinant of $A$ is $\ge0$, show that $A$ must be a rotation of $bR^3$. (2) Show that all rotaions of $R^3$ forms a group. 

 

2.

 Let $G$ be the group of all rotations of $R^3$ which send a regular dodecahedron $D$, centered at the origin, to itself. Let $G$ act on vertice of $D$. Find the number of orbits, the order of the stabilizer of a particular vertex and the order of $G$. 

說明：Dodecahedron 是正十二面體，每一面都是正五邊形。Vertex ( 其複數是vertices)

            表頂點，由 三個面相交而成，如下圖。一個頂點之stabilizer，是把此頂點固定

            之所有$g\in G$所成之集合。

 

3.

 A subgroup $H$ of a group $G$ is said to be of finite index if $[G:H]$ is finite. (Note that $H$ may not be normal.) (i) If $H$ is a subgroup of finite index, show that there exists an integer $N$ such that $g^N\in H$ for all $g\in G$. (ii) Let $G$ be the group of invertible $2\times 2$ matrices over the complex numbers. Find all subgroups of $G$ which is of finite index. 

 

4. Let $R$ be the ring of all real-valued continuous functions defined on the unit disc $x^2+y^2\le1$ with the pointwise addition and multiplication. Find all maximal ideals of $R$. 

 

5.

 Let $\le$ be a partial order defined on the set $P$. A subset $C$ of $P$ is called a chain if for any $a,b\in P$, either $a\le b$ or $b\le a$. A subset $A$ of $P$ is called an antichain if for any $a,b\in P$, neither $a\le b$ nor $b\le a$. If all chains and all antichains of $P$ are finite, prove that $P$ must be finite. 

Note: A partial order $\le$ on $P$ is a binary relation $\le$ satisfying the following: (i) Reflexivity: $a\le a$ for $a\in P$. (ii) Transitivity: if $a\le b$ and $b\le c$, then $a\le c$. (iii) Antisymmetry: if $a\le b$ and $b\le a$, then $a=b$. 

 

6.

 (i) Let $R$ be a commutative ring with $1$. Let $J$ be the intersection of all maximal ideals of $R$. Prove that $a\in J$ if and only if $1-ar$ is invertible for all $r\in R$. (ii) Find the intersection of maximal ideals of the subring $R$ of rationals defined by

$$R=\{\frac{m}{2n+1}:$$ m , n are integers$$\}.$$

7. Determine the Galois group of $x^4+3x^3-3x-2=0$ over the field of rationals.

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