位数が2の体上の10次以下の既約多項式一覧

どうもこんにちは　ごててんです
Pythonで$\mathbb{F}_2$上の既約多項式で13次以下のものをすべて計算したので, 10次以下までのものをすべて載せます. $\mathbb{F}_{1024}$の手計算などにお役立てください.

それぞれ個数は

1次：2
2次：1
3次：2
4次：3
5次：6
6次：9
7次：18
8次：30
9次：56
10次：99

となっており, たしかに以下の式の総数は226個です
( 2 + 1 + 2 + 3 + 6 + 9 + 18 + 30 + 56 + 99 = 226 )

既約多項式を載せるだけでは流石にアレなので, 既約多項式の個数を導出する式を述べておきます.（証明略）

では、既約多項式をお楽しみください......

$ X \\ X + 1 \\ X^2 + X + 1 \\ X^3 + X + 1 \\ X^3 + X^2 + 1 \\ X^4 + X + 1 \\ X^4 + X^3 + 1 \\ X^4 + X^3 + X^2 + X + 1 \\ X^5 + X^2 + 1 \\ X^5 + X^3 + 1 \\ X^5 + X^3 + X^2 + X + 1 \\ X^5 + X^4 + X^2 + X + 1 \\ X^5 + X^4 + X^3 + X + 1 \\ X^5 + X^4 + X^3 + X^2 + 1 \\ X^6 + X + 1 \\ X^6 + X^3 + 1 \\ X^6 + X^4 + X^2 + X + 1 \\ X^6 + X^4 + X^3 + X + 1 \\ X^6 + X^5 + 1 \\ X^6 + X^5 + X^2 + X + 1 \\ X^6 + X^5 + X^3 + X^2 + 1 \\ X^6 + X^5 + X^4 + X + 1 \\ X^6 + X^5 + X^4 + X^2 + 1 \\ X^7 + X + 1 \\ X^7 + X^3 + 1 \\ X^7 + X^3 + X^2 + X + 1 \\ X^7 + X^4 + 1 \\ X^7 + X^4 + X^3 + X^2 + 1 \\ X^7 + X^5 + X^2 + X + 1 \\ X^7 + X^5 + X^3 + X + 1 \\ X^7 + X^5 + X^4 + X^3 + 1 \\ X^7 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^7 + X^6 + 1 \\ X^7 + X^6 + X^3 + X + 1 \\ X^7 + X^6 + X^4 + X + 1 \\ X^7 + X^6 + X^4 + X^2 + 1 \\ X^7 + X^6 + X^5 + X^2 + 1 \\ X^7 + X^6 + X^5 + X^3 + X^2 + X + 1 \\ X^7 + X^6 + X^5 + X^4 + 1 \\ X^7 + X^6 + X^5 + X^4 + X^2 + X + 1 \\ X^7 + X^6 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^8 + X^4 + X^3 + X + 1 \\ X^8 + X^4 + X^3 + X^2 + 1 \\ X^8 + X^5 + X^3 + X + 1 \\ X^8 + X^5 + X^3 + X^2 + 1 \\ X^8 + X^5 + X^4 + X^3 + 1 \\ X^8 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^8 + X^6 + X^3 + X^2 + 1 \\ X^8 + X^6 + X^4 + X^3 + X^2 + X + 1 \\ X^8 + X^6 + X^5 + X + 1 \\ X^8 + X^6 + X^5 + X^2 + 1 \\ X^8 + X^6 + X^5 + X^3 + 1 \\ X^8 + X^6 + X^5 + X^4 + 1 \\ X^8 + X^6 + X^5 + X^4 + X^2 + X + 1 \\ X^8 + X^6 + X^5 + X^4 + X^3 + X + 1 \\ X^8 + X^7 + X^2 + X + 1 \\ X^8 + X^7 + X^3 + X + 1 \\ X^8 + X^7 + X^3 + X^2 + 1 \\ X^8 + X^7 + X^4 + X^3 + X^2 + X + 1 \\ X^8 + X^7 + X^5 + X + 1 \\ X^8 + X^7 + X^5 + X^3 + 1 \\ X^8 + X^7 + X^5 + X^4 + 1 \\ X^8 + X^7 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^8 + X^7 + X^6 + X + 1 \\ X^8 + X^7 + X^6 + X^3 + X^2 + X + 1 \\ X^8 + X^7 + X^6 + X^4 + X^2 + X + 1 \\ X^8 + X^7 + X^6 + X^4 + X^3 + X^2 + 1 \\ X^8 + X^7 + X^6 + X^5 + X^2 + X + 1 \\ X^8 + X^7 + X^6 + X^5 + X^4 + X + 1 \\ X^8 + X^7 + X^6 + X^5 + X^4 + X^2 + 1 \\ X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + 1 \\ X^9 + X + 1 \\ X^9 + X^4 + 1 \\ X^9 + X^4 + X^2 + X + 1 \\ X^9 + X^4 + X^3 + X + 1 \\ X^9 + X^5 + 1 \\ X^9 + X^5 + X^3 + X^2 + 1 \\ X^9 + X^5 + X^4 + X + 1 \\ X^9 + X^6 + X^3 + X + 1 \\ X^9 + X^6 + X^4 + X^3 + 1 \\ X^9 + X^6 + X^4 + X^3 + X^2 + X + 1 \\ X^9 + X^6 + X^5 + X^2 + 1 \\ X^9 + X^6 + X^5 + X^3 + 1 \\ X^9 + X^6 + X^5 + X^3 + X^2 + X + 1 \\ X^9 + X^6 + X^5 + X^4 + X^2 + X + 1 \\ X^9 + X^6 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^9 + X^7 + X^2 + X + 1 \\ X^9 + X^7 + X^4 + X^2 + 1 \\ X^9 + X^7 + X^4 + X^3 + 1 \\ X^9 + X^7 + X^5 + X + 1 \\ X^9 + X^7 + X^5 + X^2 + 1 \\ X^9 + X^7 + X^5 + X^3 + X^2 + X + 1 \\ X^9 + X^7 + X^5 + X^4 + X^2 + X + 1 \\ X^9 + X^7 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^9 + X^7 + X^6 + X^3 + X^2 + X + 1 \\ X^9 + X^7 + X^6 + X^4 + 1 \\ X^9 + X^7 + X^6 + X^4 + X^3 + X + 1 \\ X^9 + X^7 + X^6 + X^5 + X^4 + X^2 + 1 \\ X^9 + X^7 + X^6 + X^5 + X^4 + X^3 + 1 \\ X^9 + X^8 + 1 \\ X^9 + X^8 + X^4 + X + 1 \\ X^9 + X^8 + X^4 + X^2 + 1 \\ X^9 + X^8 + X^4 + X^3 + X^2 + X + 1 \\ X^9 + X^8 + X^5 + X + 1 \\ X^9 + X^8 + X^5 + X^4 + 1 \\ X^9 + X^8 + X^5 + X^4 + X^3 + X + 1 \\ X^9 + X^8 + X^6 + X^3 + 1 \\ X^9 + X^8 + X^6 + X^3 + X^2 + X + 1 \\ X^9 + X^8 + X^6 + X^4 + X^3 + X + 1 \\ X^9 + X^8 + X^6 + X^5 + 1 \\ X^9 + X^8 + X^6 + X^5 + X^3 + X + 1 \\ X^9 + X^8 + X^6 + X^5 + X^3 + X^2 + 1 \\ X^9 + X^8 + X^6 + X^5 + X^4 + X + 1 \\ X^9 + X^8 + X^6 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^9 + X^8 + X^7 + X^2 + 1 \\ X^9 + X^8 + X^7 + X^3 + X^2 + X + 1 \\ X^9 + X^8 + X^7 + X^5 + 1 \\ X^9 + X^8 + X^7 + X^5 + X^4 + X^2 + 1 \\ X^9 + X^8 + X^7 + X^5 + X^4 + X^3 + 1 \\ X^9 + X^8 + X^7 + X^6 + X^2 + X + 1 \\ X^9 + X^8 + X^7 + X^6 + X^3 + X + 1 \\ X^9 + X^8 + X^7 + X^6 + X^3 + X^2 + 1 \\ X^9 + X^8 + X^7 + X^6 + X^4 + X^2 + 1 \\ X^9 + X^8 + X^7 + X^6 + X^4 + X^3 + 1 \\ X^9 + X^8 + X^7 + X^6 + X^5 + X + 1 \\ X^9 + X^8 + X^7 + X^6 + X^5 + X^3 + 1 \\ X^9 + X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + X + 1 \\ X^{10} + X^3 + 1 \\ X^{10} + X^3 + X^2 + X + 1 \\ X^{10} + X^4 + X^3 + X + 1 \\ X^{10} + X^4 + X^3 + X^2 + 1 \\ X^{10} + X^5 + X^2 + X + 1 \\ X^{10} + X^5 + X^3 + X^2 + 1 \\ X^{10} + X^5 + X^4 + X^2 + 1 \\ X^{10} + X^6 + X^2 + X + 1 \\ X^{10} + X^6 + X^4 + X + 1 \\ X^{10} + X^6 + X^5 + X + 1 \\ X^{10} + X^6 + X^5 + X^2 + 1 \\ X^{10} + X^6 + X^5 + X^3 + X^2 + X + 1 \\ X^{10} + X^7 + 1 \\ X^{10} + X^7 + X^3 + X + 1 \\ X^{10} + X^7 + X^4 + X^3 + 1 \\ X^{10} + X^7 + X^5 + X^3 + 1 \\ X^{10} + X^7 + X^5 + X^3 + X^2 + X + 1 \\ X^{10} + X^7 + X^6 + X^2 + 1 \\ X^{10} + X^7 + X^6 + X^3 + 1 \\ X^{10} + X^7 + X^6 + X^4 + X^2 + X + 1 \\ X^{10} + X^7 + X^6 + X^5 + X^2 + X + 1 \\ X^{10} + X^7 + X^6 + X^5 + X^3 + X^2 + 1 \\ X^{10} + X^7 + X^6 + X^5 + X^4 + X + 1 \\ X^{10} + X^7 + X^6 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^{10} + X^8 + X^3 + X + 1 \\ X^{10} + X^8 + X^3 + X^2 + 1 \\ X^{10} + X^8 + X^4 + X^3 + 1 \\ X^{10} + X^8 + X^4 + X^3 + X^2 + X + 1 \\ X^{10} + X^8 + X^5 + X + 1 \\ X^{10} + X^8 + X^5 + X^4 + 1 \\ X^{10} + X^8 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^{10} + X^8 + X^6 + X + 1 \\ X^{10} + X^8 + X^6 + X^4 + X^2 + X + 1 \\ X^{10} + X^8 + X^6 + X^5 + 1 \\ X^{10} + X^8 + X^6 + X^5 + X^2 + X + 1 \\ X^{10} + X^8 + X^6 + X^5 + X^3 + X + 1 \\ X^{10} + X^8 + X^7 + X^2 + 1 \\ X^{10} + X^8 + X^7 + X^3 + X^2 + X + 1 \\ X^{10} + X^8 + X^7 + X^4 + X^2 + X + 1 \\ X^{10} + X^8 + X^7 + X^4 + X^3 + X + 1 \\ X^{10} + X^8 + X^7 + X^5 + 1 \\ X^{10} + X^8 + X^7 + X^5 + X^3 + X + 1 \\ X^{10} + X^8 + X^7 + X^5 + X^4 + X^3 + 1 \\ X^{10} + X^8 + X^7 + X^6 + 1 \\ X^{10} + X^8 + X^7 + X^6 + X^2 + X + 1 \\ X^{10} + X^8 + X^7 + X^6 + X^5 + X^2 + 1 \\ X^{10} + X^8 + X^7 + X^6 + X^5 + X^4 + X^2 + X + 1 \\ X^{10} + X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + X + 1 \\ X^{10} + X^9 + X^4 + X + 1 \\ X^{10} + X^9 + X^4 + X^2 + 1 \\ X^{10} + X^9 + X^5 + X + 1 \\ X^{10} + X^9 + X^5 + X^2 + 1 \\ X^{10} + X^9 + X^5 + X^4 + 1 \\ X^{10} + X^9 + X^5 + X^4 + X^2 + X + 1 \\ X^{10} + X^9 + X^6 + X + 1 \\ X^{10} + X^9 + X^6 + X^3 + X^2 + X + 1 \\ X^{10} + X^9 + X^6 + X^4 + 1 \\ X^{10} + X^9 + X^6 + X^4 + X^3 + X + 1 \\ X^{10} + X^9 + X^6 + X^5 + X^4 + X^3 + 1 \\ X^{10} + X^9 + X^6 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^{10} + X^9 + X^7 + X^2 + 1 \\ X^{10} + X^9 + X^7 + X^3 + 1 \\ X^{10} + X^9 + X^7 + X^5 + X^2 + X + 1 \\ X^{10} + X^9 + X^7 + X^5 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^7 + X^5 + X^4 + X^2 + 1 \\ X^{10} + X^9 + X^7 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ X^{10} + X^9 + X^7 + X^6 + 1 \\ X^{10} + X^9 + X^7 + X^6 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^7 + X^6 + X^4 + X + 1 \\ X^{10} + X^9 + X^7 + X^6 + X^4 + X^3 + X^2 + X + 1 \\ X^{10} + X^9 + X^7 + X^6 + X^5 + X^4 + X^2 + X + 1 \\ X^{10} + X^9 + X^7 + X^6 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^3 + X^2 + X + 1 \\ X^{10} + X^9 + X^8 + X^4 + 1 \\ X^{10} + X^9 + X^8 + X^4 + X^2 + X + 1 \\ X^{10} + X^9 + X^8 + X^4 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^5 + 1 \\ X^{10} + X^9 + X^8 + X^5 + X^3 + X + 1 \\ X^{10} + X^9 + X^8 + X^5 + X^4 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^5 + X^4 + X^3 + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^2 + X + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^4 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^4 + X^3 + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^5 + X + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^5 + X^4 + X^3 + X + 1 \\ X^{10} + X^9 + X^8 + X^6 + X^5 + X^4 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^7 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^2 + X + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^3 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^4 + X + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^5 + X^3 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^5 + X^4 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^2 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^4 + X^3 + X + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^5 + X^3 + X + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^5 + X^4 + X + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + 1 \\ X^{10} + X^9 + X^8 + X^7 + X^6 + X^5 + X^4 + X^3 + X^2 + X + 1 \\ $

ありがとうございました！！！！！！！！！！！！！！！！！！！！！！！！！

追記：
上の式をすべて掛けると（たぶん）以下の式になります（次数が合っているので大丈夫だと思いますが）

$\frac{( X^{1024} - X )( X^{512} - X )( X^{256} - X )( X^{128} - X )( X^{64} - X ) }{ ( X^8 - X )( X^4 - X )( X^4 - X )( X^2 - X ) }$

実質的にこの記事は この式の因数分解をしたことになります（）