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: 鐃緒申鐃緒申文鐃緒申砲弔鐃緒申鐃�...

鐃緒申鐃緒申鐃�II 鐃緒申鐃緒申 No.4

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{1変数多項式環とその剰余環(2), ${\Bbb F}_p$\ の単純拡大体}

鐃処週鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃縦ってわ申鐃殉ってわ申鐃緒申鐃緒申

鐃緒申鐃緒申 4.1 (鐃緒申鐃緒申3.2 鐃夙わ申鐃淑わ申)   鐃緒申 $ k$ 鐃緒申鐃渋随申犲� $ f,g\in k[X]\setminus\{0\}$ 鐃緒申鐃出わ申鐃銃￥申鐃緒申鐃塾よう鐃緒申 多鐃準式 $ a,b,d \in k[X]$ 鐃緒申存鐃淳わ申鐃暑。
  1. $ d$ 鐃緒申 $ f,g$ 鐃塾醐申鐃緒申鐃緒申任鐃緒申襦�(鐃緒申鐃淑わ申鐃緒申鐃� $ f,g \in d k[X]$)
  2. $ af +bg=d$.

鐃緒申鐃塾系わ申有鐃緒申鐃緒申(鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申)鐃緒申鐃緒申櫃隆鐃緒申椶任鐃緒申襦�

鐃緒申 4.1   $ k[X]$ 鐃塾醐申 $ p(X)$ 鐃緒申鐃緒申鐃緒申(鐃縦まり、$ p(X)$ 鐃緒申鐃緒申鐃緒申鐃� $ p(X)$ 鐃緒申鐃夙わ申鐃緒申鐃緒申椶鐃緒申鐃緒申鐃緒申 鐃祝限わ申)鐃淑わ申弌鐃� $ k[X]/p(X)k[X]$ 鐃緒申鐃塾でわ申鐃暑。

$ k$ 鐃緒申有鐃緒申鐃緒申 (鐃純え鐃出￥申 $ {\Bbb F}_p$) 鐃淑わ申仂鐃緒申鐃緒申鐃緒申里茲�鐃祝わ申鐃銃削申鐃曙た鐃塾わ申必鐃緒申的鐃緒申有鐃緒申鐃塾になる。

$ K=k[X]/p(X)k[X]$ 鐃叔わ申 $ X$ 鐃塾ワ申鐃初ス鐃緒申 $ \xi$ 鐃夙書くと￥申$ K$ 鐃緒申 $ k$ 鐃緒申 $ \xi$ 鐃夙わ申鐃緒申鐃緒申弔慮鐃緒申鐃緒申佞鐃緒申辰鐃緒申鐃緒申里砲覆辰討鐃緒申襦ｏ申鐃緒申里茲�鐃緒申鐃塾わ申 $ k$ 鐃緒申単鐃緒申鐃緒申鐃緒申鐃� 鐃夙呼ぶ￥申

$ k[X]$ 鐃塾器申鐃藷元わ申発鐃緒申鐃緒申鐃緒申鐃緒申法鐃祝つわ申鐃銃はわ申鐃夙わ申鐃夙述べわ申予鐃緒申任鐃緒申襪�鐃緒申 鐃緒申蠅�鐃緒申鐃緒申鐃緒申鐃塾わ申鐃夙わ申鐃初い鐃熟とりあ鐃緒申鐃緒申鐃塾ってわ申鐃緒申鐃夙よい鐃緒申鐃緒申鐃緒申鐃緒申

鐃緒申鐃緒申 4.1   $ k[X]$ 鐃塾醐申 $ f$ 鐃塾種申鐃緒申鐃緒申 $ 2$ 鐃殉わ申鐃緒申 $ 3$ 鐃叔わ申鐃所、 鐃緒申鐃緒申 $ f(a)=0$ 鐃夙なわ申茲�鐃緒申 $ a\in k$ 鐃緒申存鐃淳わ申鐃淑わ申鐃淑わ申弌鐃� $ f$ 鐃熟器申鐃緒申任鐃緒申襦�

鐃緒申鐃緒申鐃淑鰹申鐃緒申, 有鐃緒申鐃塾の演算鐃塾実際につわ申鐃緒申鐃塾ってわ申鐃緒申鐃塾も悪鐃緒申鐃熟なわ申鐃緒申鐃緒申鐃緒申 鐃獣￥申鐃緒申鐃緒申鐃処算鐃祝つわ申鐃銃はわ申鐃緒申覆鐃緒申颪靴鐃緒申呂覆鐃緒申隼廚鐃緒申鐃塾で￥申 鐃緒申鐃緒申鐃叔わ申 0鐃淑鰹申鐃塾醐申鐃塾逆醐申鐃塾計誌申法鐃祝つわ申鐃銃器申単鐃祝随申鐃緒申討鐃緒申鐃緒申鐃� 鐃竣はユー鐃緒申鐃緒申奪匹慮濬鐃祝￥申任鐃緒申辰董鐃緒申鐃緒申鐃緒申I 鐃叔わ申鐃叔わ申鐃旬にわ申鐃銃わ申鐃緒申呂鐃緒申任鐃緒申襦� 鐃殉わ申鐃醇ー鐃緒申鐃緒申奪匹慮濬鐃祝￥申隆鐃獣縁申鐃緒申鐃緒申鐃緒申鐃緒申蕁�

鐃緒申鐃緒申 4.1 (鐃醇ー鐃緒申鐃緒申奪匹慮濬鐃祝�)   鐃緒申鐃緒申

$\displaystyle 72l+56m=8 $

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申 $ l,m$ 鐃緒申鐃夙わ申鐃緒申筏鐃緒申茵�

(鐃緒申鐃緒申) 鐃殉わ申鐃緒申鐃塾よう鐃淑計誌申鐃緒申圓覆鐃�

  $\displaystyle 72$   $\displaystyle \div$ $\displaystyle 56$ $\displaystyle = 1$    余鐃緒申  % latex2html id marker 925 $\displaystyle 16 \qquad\qquad$ $\displaystyle 72=$ $\displaystyle 56$ $\displaystyle \times 1 +$ $\displaystyle 16$ % latex2html id marker 930 $\displaystyle \qquad\qquad \begin{pmatrix}72\\ 56 \e... ...\begin{pmatrix}1 & 1\\ 1 & 0 \end{pmatrix} \begin{pmatrix}56\\ 16 \end{pmatrix}$    
  $\displaystyle 56$   $\displaystyle \div$ $\displaystyle 16$ $\displaystyle = 3$    余鐃緒申  % latex2html id marker 935 $\displaystyle 8 \qquad\qquad$ $\displaystyle 56=$ $\displaystyle 16$ $\displaystyle \times 3 +$ $\displaystyle 8$ % latex2html id marker 940 $\displaystyle \qquad\qquad \begin{pmatrix}56\\ 16 \e... ... \begin{pmatrix}3 & 1\\ 1 & 0 \end{pmatrix} \begin{pmatrix}16\\ 8 \end{pmatrix}$    
  $\displaystyle 16$   $\displaystyle \div$ $\displaystyle 8$ $\displaystyle = 2$    余鐃緒申  % latex2html id marker 945 $\displaystyle 0 \qquad\qquad$ $\displaystyle 16=$ $\displaystyle 8$ $\displaystyle \times 2 +$ 0 % latex2html id marker 949 $\displaystyle \qquad\qquad \begin{pmatrix}16\\ 8 \en... ...= \begin{pmatrix}2 & 1\\ 1 & 0 \end{pmatrix} \begin{pmatrix}8\\ 0 \end{pmatrix}$    

鐃銃￥申鐃塾行の刻申鐃藷算わ申鐃夙み刻申錣誌申鐃夙￥申

$\displaystyle \begin{pmatrix} 72\\ 56 \end{pmatrix}= \begin{pmatrix} 1 & 1\\ ... ...egin{pmatrix} 2 & 1\\ 1 & 0 \end{pmatrix}\begin{pmatrix} 8 \\ 0 \end{pmatrix}$

鐃緒申鐃緒申鐃暑。鐃緒申鐃塾種申鐃塾縁申鐃春に醐申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃熟わ申鐃駿わ申 $ M_2({\mbox{${\Bbb Z}$}})$ 鐃塾醐申鐃夙わ申鐃緒申 鐃縦逆でわ申鐃暑こ鐃夙わ申鐃緒申鐃春わ申鐃銃￥申鐃緒申亮鐃緒申鮗，里茲�鐃緒申鐃術件申鐃緒申鐃暑こ鐃夙わ申鐃緒申鐃緒申襦�

$\displaystyle \begin{pmatrix}8 \\ 0 \end{pmatrix}$ $\displaystyle = \begin{pmatrix}2 & 1\\ 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix}... ...n{pmatrix}1 & 1\\ 1 & 0 \end{pmatrix}^{-1} \begin{pmatrix}72\\ 56 \end{pmatrix}$    
  $\displaystyle = \begin{pmatrix}0 & 1\\ 1 & -2 \end{pmatrix} \begin{pmatrix}0 & ... ...gin{pmatrix}-3 & 4 \\ 7 & 9 \end{pmatrix} \begin{pmatrix}72 \\ 56 \end{pmatrix}$    

鐃緒申鐃塾種申鐃緒申鐃緒申鐃峻わ申鐃緒申鐃旬わ申鐃緒申函鐃� $ 8=(-3)\times 72+ 4\times 56 $ 鐃緒申鐃緒申鐃暑。

(鐃緒申鐃緒申)     $ l=-3,m=4$.

鐃緒申鐃緒申 4.2   $ {\Bbb F}_{31}$ 鐃叔わ申 $ 11$ 鐃塾逆醐申鐃緒申鐃緒申茵�

(鐃緒申鐃緒申) 鐃緒申鐃緒申鐃緒申鐃�4.1鐃緒申鐃緒申鐃塾わ申 $ 31 l+11 m=1$ 鐃淑わ申 $ l,m\in {\mbox{${\Bbb Z}$}}$ 鐃緒申鐃緒申襪鰹申箸砲鐃所、

$\displaystyle 31 \times 5 +11\times (-14)=1 $

鐃緒申鐃緒申鐃暑。鐃緒申鐃塾種申鐃緒申両鐃春わ申 $ {\Bbb F}_{31}$ 鐃塾なわ申鐃叔考わ申鐃緒申函鐃�

$\displaystyle [11]_{31} [-14]_{31}=[1]_{31} $

鐃緒申鐃淑わ申鐃緒申鐃�$ 11$ 鐃緒申 $ {\Bbb F}_{31}$ 鐃叔の逆醐申鐃緒申 $ -14(=17)$ 鐃叔わ申鐃暑。

鐃緒申鐃緒申 4.1   $ K={\Bbb F}_3[X]/(X^3-X-1){\Bbb F}_3[X]$ 鐃叔わ申 $ X$ 鐃塾ワ申鐃初ス鐃緒申 $ \xi$ 鐃夙書き￥申 $ a,b\in K$ 鐃緒申 $ a=\xi^2+\xi+1$, $ b=\xi^2-1$ 鐃緒申鐃緒申鐃緒申鐃緒申襦ｏ申鐃緒申里箸鐃緒申鐃� $ a+b,a-b,ab, b^{-1}$ 鐃薯それぞ鐃緒申 $ \xi$ 鐃緒申2鐃緒申鐃淑駕申鐃塾種申鐃叔わ申鐃緒申錣件申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 4.2   $ K={\Bbb F}_{37}[X]/(X^3-X+2){\Bbb F}_{37}[X]$ 鐃叔わ申 $ X$ 鐃塾ワ申鐃初ス鐃緒申 $ \xi$ 鐃夙書くとわ申鐃緒申 $ K$ 鐃叔わ申 $ 12 \xi^2+5\xi+1$ 鐃塾逆醐申鐃緒申鐃緒申覆鐃緒申鐃緒申鐃� (鐃淑わ申鐃緒申鐃緒申鐃緒申 $ K$ 鐃熟実わ申鐃塾でわ申鐃緒申里鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申泙任麓鐃緒申鐃緒申覆鐃緒申討鐃処い鐃緒申)

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2002年5鐃緒申22鐃緒申