臺灣大學數學系

八十六學年度第二學期碩博士班資格考試試題

 代數

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選作六題, 但 (*)必作

*1.

An element σ in the symmetric goup $S_n$ induces a linear transformation $T_{\sigma}$ on an $n$-dimensional vector space by permuting the basis. If $\sigma=(12)(345) \in S_5$, then
$T_{\sigma}: e_1 \longmapsto e_2 \longmapsto e_1$, $e_3 \longmapsto e_4 \longmapsto e_5\longmapsto e_3$
Find the rational form and Jordan form of $T_\sigma$ over $\mathbb{C}$.

2.

Let $V$ be a vector space over $\mathbb{R}$ and $v_1, \cdots, v_s \in V$, $u \in V$ and $<u, v_i> \ge 0$ for all $i$
($<,>$ is the standard inner product). Let

$P=\{\alpha_1 v_1 + \cdots + \alpha_s v_s \, \big\vert\, \alpha_i \ge 0\},$ $Q=\{v\in P \, \big \vert \, <u,v> = 0 \}$.
Show that there exist $w_1, \cdots, w_t \in V$ such that

$$Q=\{\beta_1 w_1 + \cdots + \beta_t w_t \, \big \vert \, \beta_j \ge 0\}$$

*3.

Let $p, q$ be primes. Show that a nonabelian group $G$ of order $pq$ has a trivial center.

4.

(a) Find the number of elements of order $6$ in $S_6$.
(b) Find the number of elements of order $12$ in $S_6$.

*5.

Let $f(x,y)\in \mathbb{F}_q[x,y]$ where $\mathbb{F}_q$ is the finite field with $q$ elements. Suppose that $f(a,b)=0$ for all $a,b \in \mathbb{F}_q$. Show that $f(x,y)$ belongs to the ideal generated by $x^q -x$ and $y^q - y$.

6.

$$\mbox{Find all two sided ideals in }\left\{ \left[ \begin{ar... ...\right] \Big \vert a, b, c \in \mathbb{Z}\right\}.\hspace{15cm}$$

*7.

Find the Galois group of $x^4 - 4x^2 -1$ over $\mathbb{Q}$.

8.

Let $F(x)$ be the rational function field of one variable and σ be an automorphism. Show that $\sigma (x) = {{ax+b} \over {cx+d}}$ with $ad \ne bc$.

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