# $2$元体の$2$次拡大体

原始多項式$X^2+X+1$の根の$1$つを$i$とする.

足し算
||$0$|$1$|$i$|$i^2$|
|:-|:-|:-|:-|:-|
|$0$|$0$|$1$|$i$|$i^2$|
|$1$|$1$|$0$|$i^2$|$i$|
|$i$|$i$|$i^2$|$0$|$1$|
|$i^2$|$i^2$|$i$|$1$|$0$|

掛け算
||$0$|$1$|$i$|$i^2$|
|:-|:-|:-|:-|:-|
|$0$|$0$|$0$|$0$|$0$|
|$1$|$0$|$1$|$i$|$i^2$|
|$i$|$0$|$i$|$i^2$|$1$|
|$i^2$|$0$|$i^2$|$1$|$i$|

ちなみに${\bf Z}/4{\bf Z}$は
足し算
||$0$|$1$|$2$|$3$|
|:-|:-|:-|:-|:-|
|$0$|$0$|$1$|$2$|$3$|
|$1$|$1$|$2$|$3$|$0$|
|$2$|$2$|$3$|$0$|$1$|
|$3$|$3$|$0$|$1$|$2$|

掛け算
||$0$|$1$|$2$|$3$|
|:-|:-|:-|:-|:-|
|$0$|$0$|$0$|$0$|$0$|
|$1$|$0$|$1$|$2$|$3$|
|$2$|$0$|$2$|$0$|$2$|
|$3$|$0$|$3$|$2$|$1$|