臺灣大學數學系

九十學年度第二學期碩博士班資格考試試題

 代數

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1.

(15 points) Let ${\mathbb{N}}$ denote the set of positive integers. Let $F_i$ be a fields of characteristic $p_i>0$, for all $i\in{\mathbb{N}}$. Suppose that $p_i\not= p_j$, for all $i,j\in{\mathbb{N}}$. Let $F=\prod_{i\in{\mathbb{N}}}F_i$ and $K=\oplus_{i\in{\mathbb{N}}}F_i$.
a. Show that $K$ is an ideal of $F$. Is $K$ a prime ideal of $F$?
b. Show that there exists a maximal ideal $M$ of $F$ such that $K\subseteq M$ and $F/M$ is a field.
What is the characteristic of the field?

2.

(15 points) Show that there are no simple groups of order $1000$. Find a non-abelian group of order $1000$.

3.

(15 points) Let $A$ be a complex $n\times n$ matrix. Show that there exist complex polynomials $p(t)$ and $q(t)$ with $p(0)=q(0)=0$ with the properties such that $p(A)$ is diagonalizable, $q(A)$ is nilpotent and $A=p(A)+q(A)$.

4.

(15 points) Let $R$ be a ring with identity and let $M_n(R)$ denote the ring of all $n\times n$ matrices with entries in $R$.
a. Find the center of $M_n(R)$.
b. Let $I$ be a $2$-sided ideal of $R$. Show that $M_n(I)$ is a $2$-sided ideal of $M_n(R)$.
c. Describe all $2$-sided ideals of $M_n(R)$.

5.

(15 points) Let $\bar{A}$ be matrices over a field $K$ with entries indexed by positive integers ${\mathbb{N}}$ i.e. an element in $\bar{A}$ is a matrix $(a_{ij})$, $i,j\in{\mathbb{N}}$. Let $A$ be the subset of $\bar{A}$ consisting of those matrices that have only finitely many non-zero entries.
a. Show that $A$ is a simple algebra over $K$.
b. Show that $A$ is algebraic over $K$.

　

6.

(15 points) Let $R$ be a commutative Noetherian ring and $X$ an indeterminate. Prove that the power series ring $R[[X]]$ is Noetherian.

7.

(15 points) Let ${\mathbb{F}}_q$ denote the finite field of $q=p^m$ elements, where $p$ is a prime. Let ${\rm GL}_n({\mathbb{F}}_q)$ denote the group of non-singular $n\times n$ matrices over ${\mathbb{F}}_q$.
a. What is the order of ${\rm GL}_n({\mathbb{F}}_q)$.
b. What is the order of a $p$-sylow subgroup of ${\rm GL}_n({\mathbb{F}}_q)$? Find such a $p$-sylow subgroup.

8.

(15 points) Show that the following polynomials are irreducible over ${\mathbb{Q}}$ and find their Galois groups.
a. $x^3-3x+1$.
b. $x^3-3x-1$.

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