\ud790\ubca0\ub974\ud2b8\uc758 \uc601\uc810 \uc815\ub9ac(Nullstellensatz)\ub294 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc758 \ud655\uc7a5 \uc815\ub9ac\ub97c \uc720\ud55c\uc0dd\uc131\ud658\uc5d0 \uc801\uc6a9\ud55c \uc815\ub9ac\uc774\ub2e4. \uba3c\uc800 \uba87 \uac1c\uc758 \ubcf4\uc870\uc815\ub9ac\ub97c \ub3c4\uc785\ud55c \ud6c4 \uc601\uc810 \uc815\ub9ac\ub97c \uc99d\uba85\ud558\uc790.<\/p>\n
\uc815\ub9ac 1.<\/span> \uc99d\uba85<\/p>\n \\(\\mathfrak{M}\\)\uc774 \\(k\\)\uc758 \uadf9\ub300\uc544\uc774\ub514\uc5bc\uc774\uace0, \\(\\sigma\\)\uac00 \\(k[x]\\)\ub85c\ubd80\ud130 \\(k[x]\/\\mathfrak{M}\\)\uc73c\ub85c\uc758 \ud45c\uc900\uc900\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\sigma k[\\sigma x_1 ,\\, \\cdots ,\\, \\sigma x_n ]\\)\uc740 \\(\\sigma k\\)\uc758 \ud655\ub300\uccb4\uc774\ub2e4. \ub9cc\uc57d \uc720\ud55c\uc0dd\uc131\ud658\uc774 \uccb4\uc77c \ub54c \uc815\ub9ac\uc758 \uc9c4\uc220\uc744 \uc99d\uba85\ud560 \uc218 \uc788\ub2e4\uba74 \\(\\varphi \\circ \\sigma^{-1}\\)\ub97c \\(\\sigma k\\)\uc5d0 \uc801\uc6a9\ud558\uace0 \\(\\sigma k [ \\sigma x_1 ,\\, \\cdots ,\\, \\sigma x_n ]\\)\uc73c\ub85c\ubd80\ud130 \\(L\\)\ub85c\uc758 \uc77c\ub300\uc77c\uc778 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc73c\ub85c \ud655\uc7a5\ud568\uc73c\ub85c\uc368 \uc6d0\ud558\ub294 \uacb0\uacfc\ub97c \uc5bb\uc744 \uc218 \uc788\ub2e4.<\/p>\n \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(k[x]\\)\uac00 \uccb4\ub77c\uace0 \ud558\uc790. \uc774 \uccb4\uac00 \\(k\\) \uc704\uc5d0\uc11c \ub300\uc218\uc801\uc774\ub77c\uba74 \\(k[x]\\)\ub294 \ub300\uc218\uc801 \ud655\ub300\uccb4\uc774\ubbc0\ub85c \ub354 \uc774\uc0c1 \uc99d\uba85\ud560 \uac83\uc774 \uc5c6\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(k[x]\\)\uac00 \\(k\\) \uc704\uc5d0\uc11c \ub300\uc218\uc801\uc778\uc9c0 \uc54a\uc740 \uacbd\uc6b0\ub97c \uc99d\uba85\ud558\uc790. \\(t_1 ,\\) \\(\\cdots ,\\) \\(t_r\\)\uac00 \\(k[x]\\)\uc758 \ucd08\uc6d4\uae30\uc800\uc774\uace0 \\(r\\ge 1\\)\uc774\ub77c\uace0 \ud558\uc790. \\(\\varphi\\)\uac00 \\(k\\) \uc704\uc5d0\uc11c \ud56d\ub4f1\ud568\uc218\ub77c\uace0 \ud574\ub3c4 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\ub294\ub2e4. \uac01 \uc6d0\uc18c \\(x_1 ,\\) \\(\\cdots ,\\) \\(x_n\\)\uc740 \\(k(t_1 ,\\, \\cdots ,\\, t_r )\\) \uc704\uc5d0\uc11c \ub300\uc218\uc801\uc774\ub2e4. \uae30\uc57d\ub2e4\ud56d\uc2dd \\(\\operatorname{Irr}(x_i ,\\, k(t) ,\\, X)\\)\uc5d0 \\(k[t]\\)\uc758 \\(0\\)\uc774 \uc544\ub2cc \uc801\ub2f9\ud55c \uc6d0\uc18c\ub97c \uacf1\ud558\uc5ec \uacc4\uc218\uac00 \ubaa8\ub450 \\(k[t]\\)\uc5d0 \uc18d\ud558\ub294 \ub2e4\ud56d\uc2dd\uc744 \uc5bb\uc744 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \ub2e4\ud56d\uc2dd\ub4e4\uc758 \ucd5c\uace0\ucc28\ud56d\uc758 \uacc4\uc218\ub97c \\(a_1 (t) ,\\) \\(\\cdots ,\\) \\(a_n (t)\\)\ub77c\uace0 \ud558\uace0, \uc774 \uacc4\uc218\ub4e4\uc758 \uacf1\uc744 \ub530\ub984\uc815\ub9ac 1.<\/span> \uc99d\uba85<\/p>\n \uccb4\uc758 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc740 \ubaa8\ub450 \uadf8 \uc0c1\uc73c\ub85c\uc758 \ub3d9\ud615\uc0ac\uc0c1\uc774\uba70 \\(k\\) \uc704\uc5d0\uc11c\uc758 \uccb4 \\(k[x]\\)\ub85c\ubd80\ud130 \\(k\\)\uc758 \ub300\uc218\uc801 \ud3d0\ud3ec\ub85c\uc758 \uc77c\ub300\uc77c\uc778 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc774 \uc874\uc7ac\ud55c\ub2e4. \ub530\ub984\uc815\ub9ac 2.<\/span> \uc99d\uba85<\/p>\n \ud658 \\[k[x_1 ,\\, \\cdots ,\\, x_n ,\\, y_1^{-1} ,\\, \\cdots ,\\, y_m^{-1}]\\]\uc5d0 \uc815\ub9ac 1\uc744 \uc801\uc6a9\ud558\uba74 \ubc14\ub77c\ub294 \uacb0\ub860\uc744 \uc5bb\ub294\ub2e4. \\(S\\)\uac00 \\(n\\)\uac1c\uc758 \ubcc0\uc218\ub97c \uac00\uc9c4 \ub2e4\ud56d\uc2dd\ub4e4\uc758 \ud658 \\(k[X_1 ,\\, \\cdots ,\\, X_n ]\\)\uc5d0 \uc18d\ud558\ub294 \ub2e4\ud56d\uc2dd\ub4e4\uc758 \uc9d1\ud569\uc774\ub77c\uace0 \ud558\uace0 \\(L\\)\uc774 \\(k\\)\uc758 \ud655\ub300\uccb4\ub77c\uace0 \ud558\uc790. \\(L\\)\uc5d0\uc11c \\(S\\)\uc758 \uc601\uc810<\/span>(zero)\uc774\ub77c \ud568\uc740 \uc784\uc758\uc758 \\(f\\in S\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ub9ac 2.<\/span> \uc99d\uba85<\/p>\n \\(\\mathfrak{a} \\ne k[X]\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\mathfrak{a}\\)\ub97c \ud3ec\ud568\ud558\ub294 \uadf9\ub300\uc544\uc774\ub514\uc5bc \\(\\mathfrak{m}\\)\uc774 \uc874\uc7ac\ud55c\ub2e4. \\(k[X] \/ \\mathfrak{m}\\)\uc740 \\(k\\)\ub85c\ubd80\ud130 \uc720\ud55c\uc0dd\uc131\ub41c \uccb4\uc774\ub2e4. \uc989 \\(k[X] \/ \\mathfrak{m}\\)\uc740 \ubc95 \\(\\mathfrak{m}\\)\uc5d0 \ub300\ud558\uc5ec \uc815\ub9ac 2\uc5d0 \uc758\ud558\uba74 \ub2e4\ud56d\uc2dd\ub4e4\uc774 \ud558\ub098\uc758 \uccb4 \uc704\uc5d0\uc11c \uacf5\ud1b5\uc601\uc810\uc744 \uac00\uc9c0\uba74 \uc18c\uccb4 \uc704\uc5d0\uc11c \uc774\ub4e4 \ub2e4\ud56d\uc2dd\ub4e4\uc758 \uacc4\uc218\uc5d0 \uc758\ud574 \uc0dd\uc131\ub41c \uccb4\uc758 \ub300\uc218\uc801 \ud3d0\ud3ec\uc5d0 \uacf5\ud1b5\uc601\uc810\uc774 \uc874\uc7ac\ud55c\ub2e4.<\/p>\n \uc815\ub9ac 3. (\ud790\ubca0\ub974\ud2b8\uc758 \uc601\uc810 \uc815\ub9ac)<\/span><\/p>\n \\(\\mathfrak{a}\\)\uac00 \\(k[X]\\)\uc758 \uc544\uc774\ub514\uc5bc\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(f\\)\uac00 \\(k[X]\\)\uc758 \ub2e4\ud56d\uc2dd\uc774\uace0 \\(k^{\\mathrm{a}}\\)\uc5d0\uc11c \\(\\mathfrak{a}\\)\uc758 \ubaa8\ub4e0 \uc601\uc810 \\((c) = (c_1 ,\\, \\cdots ,\\, c_n )\\)\uc5d0 \ub300\ud558\uc5ec \\(f(c)=0\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f^{m} \\in \\mathfrak{a}\\)\uc778 \uc790\uc5f0\uc218 \\(m\\)\uc774 \uc874\uc7ac\ud55c\ub2e4.\n<\/p>\n<\/div>\n
\n\\(k\\)\uac00 \uccb4\uc774\uace0 \\(k[x] = k[x_1 ,\\, \\cdots ,\\, x_n ]\\)\uc774 \\(k\\) \uc704\uc5d0\uc11c\uc758 \uc720\ud55c\uc0dd\uc131\ud658\uc774\ub77c\uace0 \ud558\uc790. \ub610\ud55c \\(\\varphi : k \\,\\to\\,L\\)\uc774 \\(k\\)\ub85c\ubd80\ud130 \ub300\uc218\uc801\uc73c\ub85c \ub2eb\ud78c \uccb4 \\(L\\)\ub85c\uc758 \ub9e4\uc7a5\ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\varphi\\)\ub97c \ud655\uc7a5\ud558\uc5ec \\(k[x]\\)\ub85c\ubd80\ud130 \\(L\\)\ub85c\uc758 \uc77c\ub300\uc77c\uc778 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc744 \ub9cc\ub4e4 \uc218 \uc788\ub2e4.\n<\/p>\n<\/div>\n
\n\\[a(t) = a_1 (t) \\cdots a_n (t)\\]
\n\ub77c\uace0 \ud558\uc790. \\(a(t) \\ne 0\\)\uc774\ubbc0\ub85c \\(t_1 ‘,\\) \\(\\cdots,\\) \\(t_r ‘ \\in k^{\\mathrm{a}}\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(a(t ‘ ) \\ne 0\\)\uc774\uace0, \uac01 \\(i\\)\uc5d0 \ub300\ud558\uc5ec \\(a_i (t ‘ ) \\ne 0\\)\uc774\ub2e4. \uac01 \\(x_i\\)\uc740 \ud658
\n\\[k\\left[ t_1 ,\\, \\cdots ,\\,t_r ,\\, \\frac{1}{a_1 (t)} ,\\, \\cdots ,\\, \\frac{1}{a_r (t)} \\right]\\]
\n\uc704\uc5d0\uc11c \uc815\uc218\uc801 \uc6d0\uc18c(integral element)\uc774\ub2e4. \\(\\varphi (t_j ) = t_j ‘\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\uace0 \\(k\\) \uc704\uc5d0\uc11c \ud56d\ub4f1\ud568\uc218\uc778 \uc900\ub3d9\ud615\uc0ac\uc0c1
\n\\[\\varphi : k [ t_1 ,\\, \\cdots ,\\, t_r ] \\,\\to\\, k^{\\mathrm{a}}\\]
\n\uc744 \uc0dd\uac01\ud558\uc790. \\(\\varphi\\)\uc758 \ud575\uc744 \\(p\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(a(t) \\notin p\\)\uc774\ub2e4. \uc900\ub3d9\ud615\uc0ac\uc0c1 \\(\\varphi\\)\ub294 \uad6d\uc18c\ud658 \\(k[t]_p\\)\ub85c \ud655\uc7a5\ub418\uba70, \ud655\uc7a5\ud568\uc218\ub294 \uc720\uc77c\ud558\uac8c \uacb0\uc815\ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(\\varphi\\)\ub294
\n\\[k[t]_p [x_1 ,\\, \\cdots ,\\, x_n ]\\]
\n\uc73c\ub85c\ubd80\ud130 \\(k^{\\mathrm{a}}\\)\uc73c\ub85c\uc758 \uc77c\ub300\uc77c\uc778 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc73c\ub85c \ud655\uc7a5\ub41c\ub2e4.
\n<\/span><\/p>\n<\/div>\n
\n\\(k\\)\uac00 \uccb4\uc774\uace0 \\(k[x_1 ,\\, \\cdots ,\\, x_n ]\\)\uc774 \\(k\\) \uc704\uc5d0\uc11c\uc758 \uc720\ud55c\uc0dd\uc131\ud658\uc774\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(k[x]\\)\uac00 \uccb4\uc774\uba74 \\(k[x]\\)\ub294 \\(k\\) \uc704\uc5d0\uc11c \ub300\uc218\uc801\uc774\ub2e4.\n<\/p>\n<\/div>\n
\n<\/span><\/p>\n<\/div>\n
\n\\(k[x_1 ,\\, \\cdots ,\\, x_n ]\\)\uc774 \\(k\\) \uc704\uc5d0\uc11c\uc758 \uc720\ud55c\uc0dd\uc131\ud658\uc774\uace0 \uc774 \ud658\uc5d0\uc11c \\(0 \\ne 1\\)\uc774\uba70 \\(y_1 ,\\) \\(\\cdots ,\\) \\(y_m\\)\uc774 \uc774 \ud658\uc758 \\(0\\)\uc774 \uc544\ub2cc \uc6d0\uc18c\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(k\\) \uc704\uc5d0\uc11c\uc758 \uc900\ub3d9\ud615\uc0ac\uc0c1
\n\\[\\psi : k [x] \\,\\to\\,k^{\\mathrm{a}}\\]
\n\uc774 \uc874\uc7ac\ud558\uc5ec \uc784\uc758\uc758 \\(j=1 ,\\, \\cdots ,\\, m\\)\uc5d0 \ub300\ud558\uc5ec \\(\\psi ( y_j ) \\ne 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n<\/p>\n<\/div>\n
\n<\/span><\/p>\n<\/div>\n
\n\\[f(c_1 ,\\, \\cdots ,\\, c_n ) =0\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(L\\)\uc5d0\uc11c\uc758 \\(n\\)-\uc21c\uc11c\uc30d \\((c_1 ,\\, \\cdots ,\\, c_n )\\)\uc744 \uac00\ub9ac\ud0a8\ub2e4. \ub9cc\uc57d \\(S\\)\uac00 \ub2e8 \ud558\ub098\uc758 \ub2e4\ud56d\uc2dd \\(f\\)\ub9cc\uc744 \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4\uba74 \\((c)\\)\ub97c \\(f\\)\uc758 \uc601\uc810\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(S\\)\uc758 \ubaa8\ub4e0 \uc601\uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(L\\)\uc5d0\uc11c \\(S\\)\uc758 \ub300\uc218\uc801 \uc9d1\ud569<\/span>(algebraic set)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\mathfrak{a}\\)\uac00 \\(S\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\uc5d0 \uc758\ud574 \uc0dd\uc131\ub41c \uc544\uc774\ub514\uc5bc\uc774\ub77c\uace0 \ud558\uc790. \\(S\\subseteq \\mathfrak{a}\\)\uc774\ubbc0\ub85c \\(\\mathfrak{a}\\)\uc758 \ubaa8\ub4e0 \uc601\uc810\uc740 \\(S\\)\uc758 \uc601\uc810\uc774\ub2e4. \ub354\uc6b1\uc774 \uc774 \uba85\uc81c\uc758 \uc5ed \ub610\ud55c \ucc38\uc774\ub2e4. \uc989 \\(S\\)\uc758 \ubaa8\ub4e0 \uc601\uc810\uc740 \\(\\mathfrak{a}\\)\uc758 \uc601\uc810\uc774\ub2e4. \uc65c\ub0d0\ud558\uba74 \\(\\mathfrak{a}\\)\uc758 \ubaa8\ub4e0 \uc6d0\uc18c\ub294 \\(f_i \\in S\\)\uc640 \\(g_i \\in k[X]\\)\ub85c \uad6c\uc131\ub41c \uacb0\ud569
\n\\[g_1 (X) f_1 (X) + \\cdots + g_m (X) f_m (X)\\]
\n\ub85c \ud45c\ud604\ub418\uae30 \ub54c\ubb38\uc774\ub2e4. \ub530\ub77c\uc11c \uc9d1\ud569 \\(S\\)\uc758 \uc601\uc810\uc5d0 \ub300\ud558\uc5ec \ub17c\ud558\ub294 \uac83\uc740 \uc544\uc774\ub514\uc5bc \\(\\mathfrak{a}\\)\uc758 \uc601\uc810\uc5d0 \ub300\ud558\uc5ec \ub17c\ud558\ub294 \uac83\uacfc \uac19\ub2e4. \uc784\uc758\uc758 \uc544\uc774\ub514\uc5bc\uc740 \uc720\ud55c\uc0dd\uc131\ub418\ubbc0\ub85c \uc784\uc758\uc758 \ub300\uc218\uc801 \uc9d1\ud569\uc740 \uc720\ud55c \uac1c\uc758 \ub2e4\ud56d\uc2dd\uc758 \uc601\uc810\ub4e4\uc758 \uc9d1\ud569\uc774\ub2e4.<\/p>\n
\n\\(\\mathfrak{a}\\)\uac00 \\(k[x] = k[X_1 ,\\, \\cdots ,\\, X_n ]\\)\uc758 \uc544\uc774\ub514\uc5bc\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\mathfrak{a} = k[X]\\)\uc774\uac70\ub098 \ub610\ub294 \\(\\mathfrak{a}\\)\ub294 \\(k^{\\mathrm{a}}\\)\uc5d0\uc11c \uc601\uc810\uc744 \uac00\uc9c4\ub2e4.\n<\/p>\n<\/div>\n
\n\\[X_1 ,\\, \\cdots ,\\, X_n\\]
\n\uc758 \uc0c1\uc5d0 \uc758\ud574 \uc0dd\uc131\ub41c \uccb4\uc774\ub2e4. \uc774 \uccb4\ub294 \\(k\\) \uc704\uc5d0\uc11c \ub300\uc218\uc801\uc774\uba70 \ub300\uc218\uc801 \ud3d0\ud3ec \\(k^{\\mathrm{a}}\\)\uc758 \ubd80\ubd84\uacfc \ub3d9\ud615\uc774\ub2e4. \ubc95 \\(\\mathfrak{m}\\)\uc5d0 \ub300\ud558\uc5ec \ud45c\uc900\uc900\ub3d9\ud615\uc0ac\uc0c1\ub4e4\uc744 \ud569\uc131\ud558\uba74 \\(k[X]\\) \uc704\uc5d0\uc11c\uc758 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc744 \uc5bb\uc73c\uba70, \uc774 \uc900\ub3d9\ud615\uc0ac\uc0c1\uc5d0 \uc758\ud574 \uc5bb\uc5b4\uc9c0\ub294 \ub3d9\ud615\ubd80\ubd84\uacf5\uac04\uc5d0 \\(\\mathfrak{a}\\)\uc758 \uc601\uc810\uc774 \uc874\uc7ac\ud55c\ub2e4.
\n<\/span><\/p>\n<\/div>\n