臺灣大學數學系

八十九學年度第二學期碩博士班資格考試試題

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選作6題. $\mathbb{Z}$: integers; $\mathbb{R}$: real numbers; $GL_n(\mathbb{R})$: the group of invertible matrices; $SL_n(\mathbb{R})$:the group of matrices with determinant 1.

1.

Consider a system of linear equations,

$$\begin{eqnarray*} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n &=& b_1, \\ \cdots & & \\ a_{n1} x_1 + a_{n2} x_2 + \cdots + a_{nn} x_n &=& b_n. \end{eqnarray*}$$

$a_{ij}, b_k \in \mathbb{Z}$. Prove or disprove:

(1)

This system has a rational solution if det$A \neq 0$.

(2)

If the system has a rational solution, then it also has an integer solution.

2.

Let $\phi: F^n \rightarrow F^m$ be left multiplication: $X \mapsto AX, A \in M_{m \times n} (F)$. Prove the following are equivalent:

(1)

$A$ has a right inverse $B$ such that $AB=I$.

(2)

$\phi$ is surjective.

(3)

There is an $m \times m$ minor of $A$ whose determinant is not zero.

3.

Let $G$ be a group containing cyclic normal subgroups of order 10 and 21 respectively. Prove that $G$ contains a cyclic normal subgroup of order 210.

4.

Prove that the matrices $\left [ \begin{array}{cc} 1& 1\\ 0&1 \end{array} \right], \left [ \begin{array}{cc} 1&0 \\ 1 & 1 \end{array} \right ]$ are conjugate elements in the group $GL_n(\mathbb{R})$, but that they are not conjugate when regarded as elements of $SL_n(\mathbb{R})$.

5.

Determine all ideals of the ring $\mathbb{R}[[t]]$ of formal power series with real coefficients.

6.

Let $p$ be a prime, and $A \in M_n(\mathbb{Z})$ such that $A^p = I$, but $A \neq I$. Prove that $n \ge p -1$.

7.

Let $a,b$ be complex numbers of degree 3 over $\mathbb{Q}$, and let $K = \mathbb{Q}(a,b)$. Determine the possibilities for $[K: \mathbb{Q}]$.

8.

Let $G$ be a finite group. Prove that there exist a field $F$ and a Galois extension $K$ of $F$ whose Galois group is $G$.

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