台灣大學數學系

九十二學年度第二學期博士班資格考試題

代數

May 8, 2004

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Do $6$ out of the following $8$ problems. Do at least one problem in each section. If you do more than $6$ problems, indicate which you want graded.
We use the follow notations:
$\mathbb{R}$: the field of real numbers.
$\mathbb{Q}$: the field of rational numbers.
$\mathbb{Z}$: the ring of integers.
$\mathbb{F}_q$: the finite field of $q$ elements (where $q=p^n$ is a prime power).
$C_n$: cyclic group of order $n$.

1. Groups

(1)

Classify groups of order $20$ up to isomorphism.

(2)

Is there any isomorphism between any two of the following groups? If there is an isomorphism, find an explicit one. If not, explain why.

(a)

$S_4$.

(b)

$S_3 \times C_4$.

(c)

$Q_8 \times C_3$.

(d)

$SL(2,\mathbb{F}_3):=\left\{\left( \begin{array}{cc} a& b \\ c &d \end{array}\right)\big\vert a,b,c,d \in \mathbb{F}_3, ad-bc=1 \right\}$

2. Rings

(1)

Let $K$ be a field and $x,y,z,w$ are indeterminates. Show that $K[x,y,z,w]/(xy-zw)$ is not a UFD.

(2)

Let $K$ be a finite extension over $\mathbb{Q}$. Let $R$ be the integral closure of $\mathbb{Z}$ in $K$. Show that $R$ is a free module over $\mathbb{Z}$ of rank $=[K:\mathbb{Q}]$.

3. Fields

(1)

Determine the Galois group of $x^5-2$

(a)

over $\mathbb{Q}$

(b)

over $\mathbb{F}_5$

(2)

Determine how many irreducible polynomials are there of degree $6$ over $\mathbb{F}_p$. And verify your answer.

4. Linear Algebra

(1)

Determine the similarity classes (=conjugacy classes) of $6 \times 6$ matrices with minimal polynomial $(x^2+1)(x-1)^2$

(a)

over $\mathbb{R}$

(b)

over $\mathbb{F}_5$

(2)

Let $V$ be an even dimensional vector space over $\mathbb{R}$ and $T: V \to V$ a linear transformation. Suppose that $T^3=I$. Show that there is a linear transformation $S: V \to V$ such that $S^2=-I, ST=TS$.

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