臺灣大學數學系

九十學年度第一學期碩博士班資格考試試題

代數

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1

Let $M_{n\times n}({\mathbb{C}})$ denote the space of complex $n\times n$ matrices. Let $A\in M_{n\times n}({\mathbb{C}})$ with $A$ diagonalizable. Consider the linear map $T_A:M_{n\times n}({\mathbb{C}})\rightarrow M_{n\times n}({\mathbb{C}})$ given by

$$T_A(X)=AX-XA,\quad X\in M_{n\times n}({\mathbb{C}}).$$

Show that $T_A$ is diagonalizable. What are the eigenvalues of $T_A$? What are the eigenvectors of $T_A$?

2

Let $A$ and $B$ be $n\times n$ matrix over ${\mathbb{R}}$. Suppose that $B$ is invertible and $A^{t}BA=B$. Prove that λ is an eigenvalue of $A$ if and only if $1\over\lambda$ is an eigenvalue of $A$.

3

Let $A$ be an $n\times n$ complex matrix. Let ${\rm Tr}$ denote the trace of a matrix. Suppose that ${\rm Tr}(A)={\rm Tr}(A^2)=\cdots={\rm Tr}(A^{n})=0$. Prove that $A$ is nilpotent.

4

Let $G$ be a finite group and $g\in G$. Let $C_g$ denote the conjugacy class of $g$ and $Z_g=\{x\in G\vert gx=xg\}$. a. Show that $\vert C_g\vert={\vert G\vert\over\vert Z_g\vert}$.
b. Suppose $G=S_n$ is the symmetric group in $n$ letters. Describe the conjugacy classes of $G$.
c. Suppose that $n\ge 3$ and $g=(1,2,3)$. What is $\vert C_g\vert$?

5

Let $G$ be a finite group and $H$ a subgroup of index $n$ such that $H$ contains no non-trivial normal subgroup of $G$. Prove that $G$ may be embedded into $S_n$.

6

Let $R$ be a commutative Noetherian ring and $X$ an indeterminate. Prove that $R[X]$ is Noetherian.

7

Let $R$ be a commutative ring with $1$ and $I_1,I_2,\cdots, I_n$ be ideals of $R$ such that $I_i+I_j=R$, for $i\not=j$. Prove that the canonical map $R\rightarrow\bigoplus_{i=1}^nR/I_i$ is a surjective ring homomorphism.

8

Let $F$ be a field and $K$ a field extension of $F$ such that $[K:F]=2$. Prove that if ${\rm char}F\not=2$, then $K$ is a normal field extension of $F$.

9

Compute the Galois groups of the following polynomials over ${\mathbb{Q}}$:
a. $x^3-3x+1$.
b. $x^3-3x+3$.

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