(TableOfContents) <\/p>\n
{\\(\\mathbb{Q}(\\sqrt{2})=\\{\\alpha+\\beta\\sqrt{2} \\mid \\alpha,\\beta\\in\\mathbb{Q} \\}\\) \uc774\ub2e4.} <\/p>\n
\\noindent{\\bf (a)} $\\mathbb{Q}(\\sqrt{2})$\ub294 field\uc778\uac00? <\/p>\n
field\uc778\uac00\ub97c \ud655\uc778\ud558\ub824\uba74 \uc2dc\uac04 \uc911\uc5d0 \uacf5\ubd80\ud55c field\uc758 \uc870\uac74\uc744 \ubaa8\ub450 \ud655\uc778\ud558\uc5ec \ubcf4\uc544\uc57c \ud55c\ub2e4. \uc870\uac74\uc740 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. <\/p>\n
{\ub367\uc148\uc5d0 \uad00\ud558\uc5ec} <\/p>\n
\uc784\uc758\uc758 $k$\uc5d0 \ub300\ud558\uc5ec \\(k+0=k\\). <\/p>\n
\uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4: \\(k+h=0\\) (\uc774\ub7ec\ud55c $h$\ub97c \ubcf4\ud1b5 $-k$\ub85c \ub098\ud0c0\ub0b8\ub2e4.) <\/p>\n
{\uacf1\uc148\uc5d0 \uad00\ud558\uc5ec} <\/p>\n
$1≠0$\uc774\uba70, \uc784\uc758\uc758 $k$\uc5d0 \ub300\ud558\uc5ec \\(k1=k\\). <\/p>\n
\uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4: \\(kh=1\\) (\uc774\ub7ec\ud55c $h$\ub97c \ubcf4\ud1b5 \\(k^{-1}\\) \ub610\ub294 $\\dfrac1k$\ub85c \ub098\ud0c0\ub0b8\ub2e4.) <\/p>\n
\ub367\uc148\uacfc \uacf1\uc148\uc758 \uad00\uacc4\uc5d0 \ub300\ud558\uc5ec <\/p>\n
\uc774\uc81c $\\mathbb{Q}(\\sqrt{2})$\uc5d0 \ub300\ud558\uc5ec \uc704\uc758 \uc870\uac74\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ud655\uc778\ud560 \ub54c \uc0dd\uac01\ud558\uba74 \uc88b\uc740 \uc810\uc744 \uba87 \uac00\uc9c0 \uc815\ub9ac\ud55c\ub2e4. <\/p>\n
{\ub367\uc148\uc5d0 \uad00\ud558\uc5ec} <\/p>\n
\uc7a1\uc73c\uba74 \ub41c\ub2e4. \uc2e4\uc218\ub77c\uace0 \uc0dd\uac01\ud574 \ubcf4\uba74 \ubcf4\ud1b5 \ub54c\uc758 $0$\uc744 \uc7a1\ub294 \uac83. \\(k+0=k\\) \uc784\uc744 \ud655\uc778\ud558\uba74 \ub41c\ub2e4. <\/p>\n
\uc7a1\uc73c\uba74 \ub41c\ub2e4. $k+(-k)=0$\uc784\uc744 \ud655\uc778\ud558\uba74 \ub41c\ub2e4. <\/p>\n
\uacf1\uc148\uc5d0 \uad00\ud558\uc5ec <\/p>\n
\\(\\alpha+\\beta\\sqrt{2}\\) \uaf34\uc758 $k$\uc5d0 \ub300\ud558\uc5ec $k1=k$\uc774 \uc131\ub9bd\ud568\uc744 \ud655\uc778\ud560 \uac83. <\/p>\n
$$ k^{-1}=\\frac{\\alpha}{\\alpha^2-2\\beta^2} <\/p>\n
$$ \ub77c\uace0 \uc7a1\uc73c\uba74 \ub41c\ub2e4. \uc774\uac83\uc774 $\\mathbb{Q}(\\sqrt{2})$\uc758 \uc6d0\uc18c\uc778\uac00, \uadf8\ub9ac\uace0 \uc6d0\ud558\ub294 \uc870\uac74\uc744 \ub9cc\uc871\ud558\ub294\uac00 \uacf1\ud558\uc5ec \ud655\uc778\ud558\uc5ec \ubcfc \uac83. <\/p>\n
\ub367\uc148\uacfc \uacf1\uc148\uc758 \uad00\uacc4\uc5d0 \ub300\ud558\uc5ec <\/p>\n
\\noindent{\\bf (b)} \uc815\uc218\ub97c \uacc4\uc218\ub85c \uc774\ub7ec\ud55c \uaf34\uc758 \uc218 \ub4e4\uc744 \ub9cc\ub4e4\uba74 field\uac00 \ub418\uc9c0 \ubabb\ud55c\ub2e4. \uc774\uc720\ub294 \uacf1\uc148\uc5d0 \ub300\ud55c \uc5ed\uc6d0\uc744 \ucc3e\uc744 \uc218 \uc5c6\uc744 \ub54c\uac00 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4. \uc608\ub97c \ub4e4\uae30 \uc704\ud558\uc5ec \uc5ed\uc6d0\uc744 \ucc3e\uc744 \uc218 \uc5c6\ub294 \uc6d0\uc18c\ub97c \ud558\ub098\ub9cc \ub4e4\uc790. $k=\\sqrt{2}=0+1\\sqrt{2}$\ub77c\uace0 \ud558\uc790. \\[ \\sqrt{2}(\\alpha+\\beta\\sqrt{2})=1 \\] \uc744 \uc815\ub9ac\ud558\uba74 \\(\\alpha=0\\), \\(2\\beta=1\\) \uc774 \ub418\uc5b4 $β=1\/2$\uc774 \uc544\ub2c8\uba74 \uc548 \ub41c\ub2e4. \ub530\ub77c\uc11c $\\sqrt{2}$\ub294 \uc815\uc218\ub97c \uacc4\uc218\ub85c \ud558\ub294 \ubaa8\uc591\uc758 \uacf1\uc758 \uc5ed\uc6d0\uc744 \uac00\uc9c0\uc9c0 \uc54a\ub294\ub2e4. \uc989 field\uac00 \uc544\ub2c8\ub2e4. <\/p>\n
(\ud639\uc2dc \\(\\sqrt{2}\\) \ub300\uc2e0\uc5d0 $1+\\sqrt{2}$\ub97c \uc7a1\uc73c\uba74 \uc774\uac83\uc740 \uc5ed\uc6d0\uc744 \uac00\uc9c4\ub2e4. $-1+\\sqrt{2}$\uac00 \uc5ed\uc6d0\uc784\uc744 \ud655\uc778\ud558\uc5ec \ubcf4\uc544\ub77c.) <\/p>\n<\/div>\n<\/div>\n
1\ud559\uae30 \uad50\uacfc\uc11c\ub97c \ubcfc \uac83. <\/p>\n<\/div>\n
$ξ$\uac00 \ubb34\ub9ac\uc218\uc77c \ub54c, \\(\\mathbb{Q}\\) \uc704\uc5d0\uc11c $1,ξ$\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \ubcf4\uc774\uc790. <\/p>\n
\uc720\ub9ac\uc218 $a,b$\uc5d0 \ub300\ud558\uc5ec \\[ a\\cdot 1+b\\cdot\\xi=0\\] \uc774\ub77c \ud558\uc790. $b≠0$\uc774\ub77c \uac00\uc815\ud558\uba74 \\(\\xi=-a\/b\\) \uac00 \ub418\uc5b4 $ξ$\uac00 \uc720\ub9ac\uc218\uac00 \ub418\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c $b=0$\uc774\ub2e4. \ub530\ub77c\uc11c \\(a=-b\\xi=0\\) \uc774\ub2e4. \uc989 $a=b=0$\uc774 \ub418\uc5b4\uc57c \ud55c\ub2e4. \ub530\ub77c\uc11c $1,ξ$\ub294 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4. <\/p>\n
\uc5ed\uc73c\ub85c $1,ξ$\uac00 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub77c \ud558\uba74 $ξ$\uac00 \ubb34\ub9ac\uc218\uc784\uc744 \ubcf4\uc774\uc790.(\uc870\uad50\uc120\uc0dd\ub2d8 \ub2f5\uc774 \uc633\uc74c) $ξ$\uac00 \uc720\ub9ac\uc218\ub77c \uac00\uc815\ud558\uba74, \\[(-\\xi)\\cdot 1+1\\cdot\\xi=0\\]\uc774\ubbc0\ub85c $1,ξ$\uac00 \uc77c\ucc28\uc885\uc18d\uc774 \ub418\uc5b4 \ubaa8\uc21c\uc774\ub2e4. \ub530\ub77c\uc11c $ξ$\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4. <\/p>\n<\/div>\n
\uc2a4\uce7c\ub77c\uccb4\uac00 \\(\\mathbb{R}\\), \ub610\ub294 $ \\mathbb{C}$ \ub77c\uace0 \uac00\uc815\ud558\uba74 \uc870\uad50\uc120\uc0dd\ub2d8\uc758 \ud480\uc774\uac00 \uc633\ub2e4. <\/p>\n
\uc774\ub7ec\ud55c comment\ub97c \ub2e4\ub294 \uc774\uc720\ub294 \uc774 \ubb38\uc81c\uc758 \ud480\uc774 \uacfc\uc815\uc5d0\uc11c [$2a=0$\uc774\ubbc0\ub85c $a=0$\uc774\ub2e4] \ub77c\ub294 \ub17c\ubc95\uc744 \uc0ac\uc6a9\ud558\uae30 \ub54c\ubb38\uc774\ub2e4. \uc2a4\uce7c\ub77c\uccb4\uc5d0 \ub530\ub77c\uc11c\ub294 \uc774\ub7ec\ud55c \ub17c\ubc95\uc744 \uc4f8 \uc218 \uc5c6\ub294 \uacbd\uc6b0\uac00 \uc788\ub2e4. \uc989 \uc5b4\ub5a4 \uccb4\uc5d0\uc11c \ub294 $2=1+1=0$\uc774 \ub420 \ub54c\ub3c4 \uc788\uc73c\uba70 \uc774 \ub54c\ub294 $2=0$\uc774\ubbc0\ub85c $2a=0$\uc774\uba74\uc11c\ub3c4 $a≠0$\uc77c \uc218\ub3c4 \uc788\uae30 \ub54c\ubb38\uc774\ub2e4. <\/p>\n<\/div>\n
(a) ($ ⇒$) <\/p>\n
\uc77c\ucc28\uc885\uc18d\uc774\ub77c \uac00\uc815\ud558\uba74, \\(a(\\xi_1,\\xi_2)+b(\\eta_1,\\eta_2)=(0,0)\\) \uc778 $(a,b)≠(0,0)$\uc774 \uc788\ub2e4. <\/p>\n
$a≠0$\uc774\ub77c \uac00\uc815\ud558\uc790 ($b≠0$\uc778 \uacbd\uc6b0\ub3c4 \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.) <\/p>\n
\\(\\xi_1=-(b\/a)\\eta_1\\) \uc774\uace0 \\(\\xi_2=-(b\/a)\\eta_2\\) \uc774\ubbc0\ub85c \\[\\xi_1\\eta_2+\\xi_2\\eta_1=0\\] \uc774\ub2e4. <\/p>\n
($ ⇐$) \\(\\xi_1\\eta_2-\\xi_2\\eta_1=0\\) \uc774\ub77c \uac00\uc815\ud558\uc790. $(η_1,η_2)≠(0,0)$\uc77c \ub54c\ub294 ( $ (ξ_1,ξ_2)≠(0,0)$\uc77c \ub54c\ub3c4 \ub9c8\ucc2c\uac00\uc9c0 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud558\uba74 \ub41c\ub2e4.) \\[ \\eta_2(\\xi_1,\\xi_2) +(-\\eta_1)(\\xi_2,\\eta_2)=0 \\] \uc774 \ub41c\ub2e4. \ub530\ub77c\uc11c \ub450 \ubca1\ud130\ub294 1\ucc28 \uc885\uc18d\uc774\ub2e4. <\/p>\n
(b) \uc704\uc758 \ubb38\uc81c\uc640 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc870\uad50\uc120\uc0dd\ub2d8\uc758 \ud480\uc774\ub97c \ub530\ub974\uba74 \ub41c\ub2e4. \ub2e8\uc9c0 \ud480\uc774\uc5d0\uc11c \ub098\ub204\ub294 \uac01 \ud56d\uc774 0\uc778 \uacbd\uc6b0\ub97c \ub530\ub85c \ub530\uc838 \uc904 \ud544\uc694\uac00 \uc788\ub2e4. <\/p>\n
(c) \uc870\uad50\uc120\uc0dd\ub2d8\uc758 \ud480\uc774\uac00 \uc633\uc74c.((b)\uc640 \uac19\uc774 0\uc778 \uacbd\uc6b0\ub97c \uc870\uc2ec.) <\/p>\n<\/div>\n
\uc870\uad50\uc120\uc0dd\ub2d8 \ud480\uc774\uac00 \uc633\uc74c. \uc774 \ubb38\uc81c\ub294 \uc2dc\ud5d8\uc5d0 \uc548 \ub098\uc634 <\/p>\n<\/div>\n
\ubb38\uc81c\uc758 \uac00\uc815\uc73c\ub85c\ubd80\ud130 $ V⊂ M∪ N $ \uc774\ubbc0\ub85c $ V=M∪ N $ \uc774 \ub418\uace0 \ub530\ub77c\uc11c \uc870\uad50\uc120\uc0dd\ub2d8\uc758 case (i) \uc740 \ud544\uc694 \uc5c6\ub2e4. \ub9c8\uc9c0\ub9c9\uc5d0\uc11c \ub450 \ubc88\uc9f8 \uc904, \ub9c8\uc9c0\ub9c9 \uc2dd\uc740 \\( v_1+v_2 \\not\\in N \\) \uc774\ub77c\uace0 \uc368\uc57c \uc633\ub2e4. \uc774 \uc904\uc774 \uc65c \uc131\ub9bd\ud558\ub294\uc9c0\ub97c \ud655\uc778\ud560 \uac83. <\/p>\n<\/div>\n
(a) \uc774 \ubb38\uc81c\ub97c \ud478\ub294\ub370 \ucc28\uc6d0\uc815\ub9ac\ub97c \uc4f0\ub294 \uac83\uc740 \uc633\uc9c0 \uc54a\ub2e4. \uc65c\uc778\uac00\ub294 \ucc28\uc6d0\uc815\ub9ac\ub97c \uc798 \uc0b4\ud3b4\ubcf4\uba74 \uc54c \uc218 \uc788\uc9c0\ub9cc, \uc774 \uc815\ub9ac\ub294 \uc804\uccb4\uacf5\uac04\uc774 \uc720\ud55c\ucc28\uc6d0\uc77c \ub54c \uc720\ud55c\uac1c\uc758 basis\ub97c \uad6c\ud558\ub294 \ubc29\ubc95\uc73c\ub85c \uc99d\uba85\ud558\uace0 \uc788\ub2e4. \uc6b0\ub9ac\uac00 \uc99d\uba85\ud55c \ucc28\uc6d0\uc815\ub9ac\ub294 \ucc28\uc6d0\uc774 \ubb34\ud55c\uc778 \uacf5\uac04 $ P $ \uc5d0\ub294 \uc801\uc6a9\ud560 \uc218 \uc5c6\ub294 \ud615\ud0dc\uc774\ub2e4. \uc6b0\ub9ac\uac00 \uc774 \ubb38\uc81c\ub97c \ud478\ub294 \ubc29\ubc95\uc740 \uc9c1\uc811 $ P\/M $ \uc5d0\uc11c \uc5bc\ub9c8\ub4e0\uc9c0 \ub9ce\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc778 \ubca1\ud130\ub4e4\uc744 \ucc3e\uc544\ub0b4\uc11c \ubcf4\uc5ec\uc90c\uc73c\ub85c\uc368 \uc774\ub2e4. \uc989 $ t^{n+1},…,t^{n+k} $ \uc740 $ P $ \uc758 \ubca1\ud130\ub4e4\uc774\ub2e4. \uc774 \uac01\uac01\uc758 \ubca1\ud130\ub4e4\uc744 \ud3ec\ud568\ud558\ub294 $ P\/M $ \uc758 \uc6d0\uc18c\ub4e4\uc740 \uac01\uac01 $ \\{t^{n+1}\\} $ , … , $ \\{t^{n+k}\\} $ \ub77c\uace0 \uc4f8 \uc218 \uc788\ub2e4. \uc774\uac83\ub4e4\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc744 \ubcf4\uc774\uba74 \uc77c\ucc28\ub3c5\ub9bd\uc778 \ubca1\ud130\uc758 \uac1c\uc218\uac00 \uc784\uc758\uc758 \uc790\uc5f0\uc218 $ k $ \ub9cc\ud07c \uc788\ub2e4\ub294 \uac83\uc774 \ub418\uc5b4 \uc720\ud55c\ud55c \uac1c\uc218\uc758 basis\ub97c \uac00\uc9c8 \uc218 \uc5c6\ub2e4. \uc989 \ucc28\uc6d0\uc774 \ubb34\ud55c\uc774 \ub418\ub294 \uac83\uc774\ub2e4. \uc774\uc81c \uc774\ub4e4\uc774 \uc77c\ucc28\ub3c5\ub9bd\uc784\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\[ \\{ a_1 t^{n+1} + \\cdots + a_k t^{n+k} \\} = \\{0\\} \\] \ub77c\uace0 \ud558\uba74, \\( a_1 t^{n+1} + \\cdots + a_k t^{n+k} \\) \ub294 $ M $ \uc758 \uc6d0\uc18c\ub77c\ub294 \ub73b\uc774\uace0 \ub530\ub77c\uc11c \uc801\ub2f9\ud55c $ b_i $ \ub4e4\uc5d0 \ub300\ud558\uc5ec \\[ a_1 t^{n+1} + \\cdots + a_k t^{n+k} = b_0 + b_1 t + \\cdots + b_n t^n \\] \uc774 \ub41c\ub2e4. \uc774\ub85c\ubd80\ud130 \\( a_1 = \\cdots = a_k = b_0 = \\cdots = b_n = 0 \\) \uc784\uc744 \uc54c \uc218 \uc788\ub2e4. \uc989 \uc774\ub4e4\uc740 \uc77c\ucc28\ub3c5\ub9bd\uc774\ub2e4. <\/p>\n<\/div>\n
(TableOfContents) \uccab\ubc88\uc9f8 \uc219\uc81c \ud480\uc774\uc785\ub2c8\ub2e4. \ubb38\uc81c 1 {\\(\\mathbb{Q}(\\sqrt{2})=\\{\\alpha+\\beta\\sqrt{2} \\mid \\alpha,\\beta\\in\\mathbb{Q} \\}\\) \uc774\ub2e4.} \\noindent{\\bf (a)} $\\mathbb{Q}(\\sqrt{2})$\ub294 field\uc778\uac00? field\uc778\uac00\ub97c \ud655\uc778\ud558\ub824\uba74 \uc2dc\uac04 \uc911\uc5d0 \uacf5\ubd80\ud55c field\uc758 \uc870\uac74\uc744 \ubaa8\ub450 \ud655\uc778\ud558\uc5ec \ubcf4\uc544\uc57c \ud55c\ub2e4. \uc870\uac74\uc740 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc774\ub2e4. {\ub367\uc148\uc5d0 \uad00\ud558\uc5ec} \\(k+h=h+k\\) \\(k+(h+l)=(k+h)+l\\) \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc6d0\uc18c $0$\uc774 \uc720\uc77c\ud558\uac8c \uc874\uc7ac\ud55c\ub2e4: \uc784\uc758\uc758 $k$\uc5d0 \ub300\ud558\uc5ec \\(k+0=k\\). \uc784\uc758\uc758 $k$\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc6d0\uc18c … \ub354 \uc77d\uae30<\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-3852","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"distributor_meta":false,"distributor_terms":false,"distributor_media":false,"distributor_original_site_name":"\uae40\uc601\uc6b1","distributor_original_site_url":"https:\/\/mathematicians.korea.ac.kr\/ywkim","push-errors":false,"_links":{"self":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3852","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/comments?post=3852"}],"version-history":[{"count":1,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3852\/revisions"}],"predecessor-version":[{"id":3853,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/posts\/3852\/revisions\/3853"}],"wp:attachment":[{"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/media?parent=3852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/categories?post=3852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematicians.korea.ac.kr\/ywkim\/wp-json\/wp\/v2\/tags?post=3852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}