2022 高等代数
- 证明： p ( x ) p(x) p(x) 是不可约多项式的充要条件是对任意的多项式 f ( x ) , g ( x ) f(x), g(x) f(x),g(x)，若 p ( x ) ∣ f ( x ) g ( x ) p(x) \mid f(x)g(x) p(x)∣f(x)g(x)，则有 p ( x ) ∣ f ( x ) p(x) \mid f(x) p(x)∣f(x) 或 p ( x ) ∣ g ( x ) p(x) \mid g(x) p(x)∣g(x)。
⇒ \Rightarrow ⇒
当 p ( x ) p(x) p(x) 是不可约多项式，且 p ( x ) ∣ f ( x ) g ( x ) p(x) \mid f(x)g(x) p(x)∣f(x)g(x) ，若 p ( x ) ∤ f ( x ) p(x) \nmid f(x) p(x)∤f(x)，则两多项式互素，即 ( p ( x ) , f ( x ) ) = 1 (p(x),f(x))=1 (p(x),f(x))=1，于是 p ( x ) ∣ g ( x ) p(x) \mid g(x) p(x)∣g(x)
⇐ \Leftarrow ⇐
若 p ( x ) p(x) p(x) 可约，设 p ( x ) = p 1 ( x ) p 2 ( x ) p(x)=p_1(x)p_2(x) p(x)=p1(x)p2(x) ， ∂ ( p i ( x ) ) = deg p i ( x ) < deg p ( x ) = ∂ ( p ( x ) ) ( i = 1 , 2 ) \partial \left( p_i(x) \right) = \deg p_i(x) < \deg p(x) = \partial \left( p(x) \right)\,(i=1,2) ∂(pi(x))=degpi(x)<degp(x)=∂(p(x))(i=1,2) ， p ( x ) ∣ p 1 ( x ) p 2 ( x ) p(x) \mid p_1(x)p_2(x) p(x)∣p1(x)p2(x) ，但 p ( x ) ∤ p 1 ( x ) p(x) \nmid p_1(x) p(x)∤p1(x)， p ( x ) ∤ p 2 ( x ) p(x) \nmid p_2(x) p(x)∤p2(x)
- 计算行列式
∣ 2 n − 2 2 n − 1 − 2 ⋯ 2 3 − 2 2 2 − 2 3 n − 3 3 n − 1 − 3 ⋯ 3 3 − 3 3 2 − 3 ⋮ ⋮ ⋱ ⋮ ⋮ n n − n n n − 1 − n ⋯ n 3 − n n 2 − n ∣ . \begin{vmatrix} 2^n - 2 & 2^{n-1} - 2 & \cdots & 2^3 - 2 & 2^2 - 2 \\ 3^n - 3 & 3^{n-1} - 3 & \cdots & 3^3 - 3 & 3^2 - 3 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n^n - n & n^{n-1} - n & \cdots & n^3 - n & n^2 - n \\ \end{vmatrix}. 2n−23n−3⋮nn−n2n−1−23n−1−3⋮nn−1−n⋯⋯⋱⋯23−233−3⋮n3−n22−232−3⋮n2−n .
法 1
∣ 2 n − 2 2 n − 1 − 2 ⋯ 2 2 − 2 3 n − 3 3 n − 1 − 3 ⋯ 3 2 − 3 ⋮ ⋮ ⋱ ⋮ n n − n n n − 1 − n ⋯ n 2 − n ∣ = ∣ 2 n − 1 ( 2 − 1 ) 2 n − 2 ( 2 − 1 ) ⋯ 2 ( 2 − 1 ) 3 n − 1 ( 3 − 1 ) 3 n − 2 ( 3 − 1 ) ⋯ 3 ( 3 − 1 ) ⋮ ⋮ ⋱ ⋮ n n − 1 ( n − 1 ) n n − 2 ( n − 1 ) ⋯ n ( n − 1 ) ∣ = ( n − 1 ) ! ∣ 2 n − 1 2 n − 2 ⋯ 2 3 n − 1 3 n − 2 ⋯ 3 ⋮ ⋮ ⋱ ⋮ n n − 1 n n − 2 ⋯ n ∣ = n ! ( n − 1 ) ! ∣ 2 n − 2 2 n − 3 ⋯ 1 3 n − 2 3 n − 3 ⋯ 1 ⋮ ⋮ ⋱ ⋮ n n − 2 n n − 3 ⋯ 1 ∣ = n ! ( n − 1 ) ! ( − 1 ) ( n − 1 ) ( n − 2 ) 2 ∏ 2 ≤ j < i ≤ n ( i − j ) = ( − 1 ) ( n − 1 ) ( n − 2 ) 2 ∏ k = 1 n k ! \begin{align*} \left| \begin{array}{cccc} {2^n - 2} & {2^{n - 1} - 2} & \cdots & {2^2 - 2} \\ {3^n - 3} & {3^{n - 1} - 3} & \cdots & {3^2 - 3} \\ \vdots & \vdots & \ddots & \vdots \\ {n^n - n} & {n^{n - 1} - n} & \cdots & {n^2 - n} \end{array} \right| &= \left| \begin{array}{cccc} {2^{n - 1} (2 - 1)} & {2^{n - 2} (2 - 1)} & \cdots & {2 (2 - 1)} \\ {3^{n - 1} (3 - 1)} & {3^{n - 2} (3 - 1)} & \cdots & {3 (3 - 1)} \\ \vdots & \vdots & \ddots & \vdots \\ {n^{n - 1} (n - 1)} & {n^{n - 2} (n - 1)} & \cdots & {n (n - 1)} \end{array} \right| \\ &= (n - 1)! \left| \begin{array}{cccc} {2^{n - 1}} & {2^{n - 2}} & \cdots & 2 \\ {3^{n - 1}} & {3^{n - 2}} & \cdots & 3 \\ \vdots & \vdots & \ddots & \vdots \\ {n^{n - 1}} & {n^{n - 2}} & \cdots & n \end{array} \right| \\ &= n! (n - 1)! \left| \begin{array}{cccc} {2^{n - 2}} & {2^{n - 3}} & \cdots & 1 \\ {3^{n - 2}} & {3^{n - 3}} & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ {n^{n - 2}} & {n^{n - 3}} & \cdots & 1 \end{array} \right| \\ &= n! (n - 1)! \left( -1 \right)^{\frac{(n - 1)(n - 2)}{2}} \prod_{2 \le j < i \le n} (i - j) \\ &= \left( -1 \right)^{\frac{(n - 1)(n - 2)}{2}} \prod_{k = 1}^n k! \end{align*}
2n−23n−3⋮nn−n2n−1−23n−1−3⋮nn−1−n⋯⋯⋱⋯22−232−3⋮n2−n
=
2n−1(2−1)3n−1(3−1)⋮nn−1(n−1)2n−2(2−1)3n−2(3−1)⋮nn−2(n−1)⋯⋯⋱⋯2(2−1)3(3−1)⋮n(n−1)
=(n−1)!
2n−13n−1⋮nn−12n−23n−2⋮nn−2⋯⋯⋱⋯23⋮n
=n!(n−1)!
2n−23n−2⋮nn−22n−33n−3⋮nn−3⋯⋯⋱⋯11⋮1
=n!(n−1)!(−1)2(n−1)(n−2)2≤j<i≤n∏(i−j)=(−1)2(n−1)(n−2)k=1∏nk!
法 2
∣ 2 n − 2 2 n − 1 − 2 ⋯ 2 3 − 2 2 2 − 2 3 n − 3 3 n − 1 − 3 ⋯ 3 3 − 3 3 2 − 3 ⋮ ⋮ ⋱ ⋮ ⋮ n n − n n n − 1 − n ⋯ n 3 − n n 2 − n ∣ = ∣ 1 1 1 ⋯ 1 1 0 2 n − 2 2 n − 1 − 2 ⋯ 2 3 − 2 2 2 − 2 0 3 n − 3 3 n − 1 − 3 ⋯ 3 3 − 3 3 2 − 3 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 0 n n − n n n − 1 − n ⋯ n 3 − n n 2 − n ∣ = n ! ∣ 1 1 1 ⋯ 1 1 1 2 n − 1 2 n − 2 ⋯ 2 2 2 1 3 n − 1 3 n − 2 ⋯ 3 2 3 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 1 n n − 1 n n − 2 ⋯ n 2 n ∣ = n ! ( − 1 ) ( n − 1 ) ( n − 2 ) 2 ∣ 1 1 1 ⋯ 1 1 1 2 2 2 ⋯ 2 n − 2 2 n − 1 1 3 3 2 ⋯ 3 n − 2 3 n − 1 ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ 1 n n 2 ⋯ n n − 2 n n − 1 ∣ = ( − 1 ) ( n − 1 ) ( n − 2 ) 2 ∏ k = 1 n k ! \begin{align*} \left| \begin{array}{ccccc} {2^n - 2} & {2^{n - 1} - 2} & \cdots & {2^3 - 2} & {2^2 - 2} \\ {3^n - 3} & {3^{n - 1} - 3} & \cdots & {3^3 - 3} & {3^2 - 3} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {n^n - n} & {n^{n - 1} - n} & \cdots & {n^3 - n} & {n^2 - n} \end{array} \right| &= \left| \begin{array}{cccccc} 1 & 1 & 1 & \cdots & 1 & 1 \\ 0 & {2^n - 2} & {2^{n - 1} - 2} & \cdots & {2^3 - 2} & {2^2 - 2} \\ 0 & {3^n - 3} & {3^{n - 1} - 3} & \cdots & {3^3 - 3} & {3^2 - 3} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & {n^n - n} & {n^{n - 1} - n} & \cdots & {n^3 - n} & {n^2 - n} \end{array} \right| \\ &= n! \left| \begin{array}{cccccc} 1 & 1 & 1 & \cdots & 1 & 1 \\ 1 & {2^{n - 1}} & {2^{n - 2}} & \cdots & {2^2} & 2 \\ 1 & {3^{n - 1}} & {3^{n - 2}} & \cdots & {3^2} & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & {n^{n - 1}} & {n^{n - 2}} & \cdots & {n^2} & n \end{array} \right| \\ &= n! \left( -1 \right)^{\frac{(n - 1)(n - 2)}{2}} \left| \begin{array}{cccccc} 1 & 1 & 1 & \cdots & 1 & 1 \\ 1 & 2 & {2^2} & \cdots & {2^{n - 2}} & {2^{n - 1}} \\ 1 & 3 & {3^2} & \cdots & {3^{n - 2}} & {3^{n - 1}} \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & n & {n^2} & \cdots & {n^{n - 2}} & {n^{n - 1}} \end{array} \right| \\ &= \left( -1 \right)^{\frac{(n - 1)(n - 2)}{2}} \prod_{k = 1}^n k! \end{align*}