2025-04-21 试卷

(20points).

1. (10 points). Assume that $p$ is a prime number greater than 2. Prove that the polynomial

   $$ f (x) = x^{p-1} + x^{p-2} + \cdots + x + 1 $$

   is irreducible in the rational number field $ℚ$.

2. (10 points). For an integer coefficient polynomial

   $$ f (x) = x^{n} + a_1 x^{n-1} + \cdots + a_{n-1} x + a_{n}, $$

   the sufficient and necessary condition for $f (x)$ to have (non-zero) integer roots is that there are $2(n-1)$ integers $\{ b_i, c_i , \ i = 1, \ldots, n-1\}$ such that

   1. $a_i = b_i + c_i$, for $i = 1, \cdots, n-1$; and
   2. $\frac{1}{c_1} = \frac{b_1}{c_2} = \frac{b_2}{c_3} = \cdots = \frac{b_{n-2}}{c_{n-1}} = \frac{b_{n-1}}{a_{n}}$.

(20 points).

1. (10 points). Assume that $a,b ∈ ℂ$ are two complex numbers. Prove the following two subspaces of $ℂ[x]$,

   $$ \begin{aligned} & V_a=\{f(x) \in \mathbb{C}[x] \mid f(a)=0\}, \\[6pt] & V_b=\{g(x) \in \mathbb{C}[x] \mid g(b)=0\} , \end{aligned} $$

   are isomorphic.

2. (10 points). Assume that $ε_1, ε_2, ε_3$ is a basis of the linear space $V$, and $f_1, f_2, f_3$ is the dual basis of $ε_1, ε_2, ε_3$. Let

   $$ α_1=ε_1+ε_2+ε_3,\quad {α}_2=ε_2+ε_3,\quad {α}_3=ε_3. $$

   1. Prove that $α_1, α_2, α_3$ is a basis of $V$;
   2. Find the dual basis of $α_1, α_2, α_3$ (using $f_1, f_2, f_3$ to represent).

(20 points). Suppose $V$ is an $n$-dimensional vector space over a number field, and let $V_1, V_2, \ldots, V_s$ be $s$ proper subspaces of $V$. Prove that

1. There exists some $α ∈ V$ such that $α ∉ V_1 ∪ V_2 ∪ \cdots ∪ V_s$;

2. There exists a basis of $V, ε _1, ε _2, \ldots, ε _n$, such that

   $$ \left\{ε _1, ε _2, \ldots, ε _n\right\} ⋂ \left(V_1 ∪ V_2 ∪ \cdots ∪ V_s\right)=∅. $$

(15 points). Suppose $φ _1, φ _2, \ldots, φ_m$ are linear maps over the $n$-dimensional vector space $V$, such that

$$ φ _i^2 = φ _i\quad \text{and}\quad φ _i φ _j = 0 \quad (i ≠ j, \ 1 ≤ i, \ j ≤ m). $$

Prove that

$$ V=\operatorname{im} \varphi_1 ⊕ \operatorname{im} \varphi_2 ⊕ \cdots ⊕ \operatorname{im} φ_m ⊕ \bigcap_{i=1}^m \operatorname{ker} φ_i . $$

(15 points). Suppose the linear map $𝒜 (X)=AXA^T$ over $ℝ^{n × n}$, where $A$ is an $n$-order real square matrix with $\operatorname{rank}(A) = r$. Find the dimension and a basis of $\operatorname{im} 𝒜$.

(10 points). You are encouraged to have a great amount of reading and thinking besides the classes. This problem is meant to test the your understanding. Please give a new theorem/question that has neither appeared in classes nor in our textbook, and then give a proof/solution to this theorem/question. It can be a phenomenon you have summarized, a theorem in other books, or a theorem that is beyond what we have learned.