{"id":7531,"date":"2021-11-20T13:59:06","date_gmt":"2021-11-20T06:59:06","guid":{"rendered":"https:\/\/www.vietlearn.org\/post\/?p=7531"},"modified":"2021-11-20T15:08:49","modified_gmt":"2021-11-20T08:08:49","slug":"co-phai-cac-phuong-trinh-deu-co-the-giai-bang-cong-thuc-khong","status":"publish","type":"post","link":"https:\/\/www.vietlearn.org\/kien-thuc\/co-phai-cac-phuong-trinh-deu-co-the-giai-bang-cong-thuc-khong\/","title":{"rendered":"C\u00f3 ph\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ec1u c\u00f3 th\u1ec3 gi\u1ea3i b\u1eb1ng c\u00f4ng th\u1ee9c kh\u00f4ng?"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Nhi\u1ec1u ng\u01b0\u1eddi th\u00edch d\u00f9ng c\u00f4ng th\u1ee9c khi gi\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh v\u00ec ch\u1ec9 c\u1ea7n theo c\u00e1c qu\u00e1 tr\u00ecnh v\u00e0 quy ph\u1ea1m kh\u00f4ng c\u1ea7n ph\u1ea3i t\u1ed1n nhi\u1ec1u suy ngh\u0129. V\u00ed nh\u01b0 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai ax2 + bx + c = 0 (trong \u0111\u00f3 a \u2260 0) ta ch\u1ec9 c\u1ea7n d\u00f9ng c\u00f4ng th\u1ee9c: ta c\u00f3 th\u1ec3 \u0111\u01b0a ra hai nghi\u1ec7m c\u1ee7a ph\u01b0\u01a1ng tr\u00ecnh. V\u00ec v\u1eady trong th\u1eddi gian d\u00e0i, ng\u01b0\u1eddi ta b\u1ecf c\u00f4ng t\u00ecm c\u00e1c c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh c\u00e1c b\u1eadc, v\u00e0 tr\u1edf th\u00e0nh m\u1ed9t v\u1ea5n \u0111\u1ec1 quan t\u00e2m tr\u1ecdng \u0111i\u1ec3m c\u1ee7a \u0111\u1ea1i s\u1ed1 h\u1ecdc.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">V\u00e0o n\u0103m 1535, nh\u00e0 to\u00e1n h\u1ecdc Italia l\u1ea7n \u0111\u1ea7u ti\u00ean t\u00ecm ra c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc ba. H\u1ecd t\u00ecm c\u00e1ch bi\u1ebfn \u0111\u1ed5i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc ba th\u00e0nh ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai, sau \u0111\u00f3 nh\u1edd gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ec3 t\u00ecm c\u00e1c nghi\u1ec7m. Nh\u1edd \u00fd t\u01b0\u1edfng c\u1ee7a ph\u01b0\u01a1ng ph\u00e1p n\u00e0y, v\u1ec1 sau nh\u00e0 to\u00e1n h\u1ecdc Italia Ferali \u0111\u00e3 t\u00ecm c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc 4 v\u00e0 l\u1ea7n n\u1eefa ch\u1ee9ng minh t\u00ednh h\u1eefu hi\u1ec7u c\u1ee7a \u00fd t\u01b0\u1edfng n\u00e0y.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nh\u01b0 v\u1eady c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai, b\u1eadc ba, b\u1eadc b\u1ed1n \u0111\u1ec1u c\u00f3 th\u1ec3 gi\u1ea3i qua c\u00e1c c\u00f4ng th\u1ee9c. V\u1eady v\u1edbi c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc n\u0103m, b\u1eadc s\u00e1u v\u00e0 b\u1eadc cao h\u01a1n c\u00f3 th\u1ec3 d\u00f9ng c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i \u0111\u01b0\u1ee3c kh\u00f4ng? Xu\u1ea5t ph\u00e1t t\u1eeb nh\u1eadn th\u1ee9c n\u00e0y t\u1eeb th\u1ebf k\u1ec9 XVII tr\u1edf \u0111i, c\u00e1c nh\u00e0 to\u00e1n h\u1ecdc \u0111\u1ec1u ra s\u1ee9c t\u00ecm c\u00e1c c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc n\u0103m v\u00e0 b\u1eadc cao h\u01a1n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">C\u00f3 \u0111i\u1ec1u l\u1ea1 l\u00e0 v\u00e0o th\u1ebf k\u1ec9 XVI, nh\u00e0 to\u00e1n h\u1ecdc Ferali 20 tu\u1ed5i, kh\u00f4ng t\u1ed1n nhi\u1ec1u th\u1eddi gian l\u1eafm \u0111\u00e3 t\u00ecm ra c\u00f4ng th\u1ee9c gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc b\u1ed1n, \u0111i\u1ec1u m\u00e0 trong su\u1ed1t hai th\u1ebf k\u1ec9 XVI, XVII kh\u00f4ng \u00edt nh\u00e0 to\u00e1n h\u1ecdc t\u00e0i ba \u0111\u00e3 nghi\u00ean c\u1ee9u mong t\u00ecm c\u00e1ch gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh cao h\u01a1n m\u1ed9t b\u1eadc l\u00e0 ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc n\u0103m nh\u01b0ng kh\u00f4ng t\u00ecm th\u1ea5y c\u00f4ng th\u1ee9c.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Th\u1ebf c\u00f3 ph\u1ea3i v\u1edbi c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh t\u1eeb b\u1eadc 5 tr\u1edf l\u00ean kh\u00f4ng gi\u1ea3i \u0111\u01b0\u1ee3c b\u1eb1ng c\u00f4ng th\u1ee9c? V\u1ea5n \u0111\u1ec1 n\u00e0y \u0111\u01b0\u1ee3c \u0111\u1eb7t ra v\u00e0o n\u0103m 1824. Nh\u00e0 to\u00e1n h\u1ecdc Na uy 22 tu\u1ed5i l\u00e0 Abel sau b\u1ed1n n\u0103m n\u1ed7 l\u1ef1c \u0111\u00e3 ch\u1ee9ng minh: v\u1edbi c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 b\u1eadc b\u1eb1ng 5 ho\u1eb7c l\u1edbn h\u01a1n kh\u00f4ng th\u1ec3 bi\u1ec3u di\u1ec5n c\u00e1c nghi\u1ec7m c\u1ee7a ch\u00fang qua c\u00e1c h\u1ec7 s\u1ed1 b\u1eb1ng c\u00e1c ph\u00e9p t\u00ednh s\u1ed1 h\u1ecdc c\u01a1 b\u1ea3n (c\u1ed9ng, tr\u1eeb, nh\u00e2n, chia, khai c\u0103n, lu\u1ef9 th\u1eeba..). T\u1eeb \u0111\u00f3 v\u1ea5n \u0111\u1ec1 t\u00ecm c\u00f4ng th\u1ee9c \u0111\u1ec3 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc cao t\u1eeb b\u1eadc 5 tr\u1edf \u0111i m\u1edbi k\u1ebft th\u00fac.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Th\u1ebf nh\u01b0ng l\u00ed lu\u1eadn v\u1ec1 gi\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh ch\u01b0a ch\u1ea5m d\u1ee9t.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nh\u1eefng k\u1ebft qu\u1ea3 c\u1ee7a Abel kh\u00f4ng h\u1ec1 n\u00f3i l\u00e0 kh\u00f4ng c\u00f3 c\u00f4ng th\u1ee9c \u0111\u1ec3 bi\u1ec3u di\u1ec5n c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh c\u00f3 b\u1eadc l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 5. V\u00ed d\u1ee5 v\u1edbi ph\u01b0\u01a1ng tr\u00ecnh x5 = N, v\u1edbi ph\u01b0\u01a1ng tr\u00ecnh \u0111\u01a1n gi\u1ea3n n\u00e0y ta c\u00f3 th\u1ec3 t\u00ednh tr\u1ef1c ti\u1ebfp nghi\u1ec7m b\u1eb1ng ph\u00e9p to\u00e1n khai c\u0103n. Do \u0111\u00f3 v\u1ea5n \u0111\u1ec1 \u0111\u01b0\u1ee3c \u0111\u1ea9y l\u00ean m\u1ed9t b\u01b0\u1edbc m\u1edbi. C\u00e1c nh\u00e0 to\u00e1n h\u1ecdc \u0111\u01b0a ra lu\u1eadn \u0111\u1ec1 v\u1edbi ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc cao ph\u1ea3i c\u00f3 d\u1ea1ng nh\u01b0 th\u1ebf n\u00e0o th\u00ec c\u00f3 th\u1ec3 bi\u1ec3u di\u1ec5n nghi\u1ec7m qua c\u00e1c h\u1ec7 s\u1ed1 ph\u01b0\u01a1ng tr\u00ecnh th\u00f4ng qua c\u00e1c ph\u00e9p to\u00e1n s\u1ed1 h\u1ecdc? So v\u1edbi lu\u1eadn \u0111\u1ec1 tr\u01b0\u1edbc \u0111\u00e2y, v\u1ea5n \u0111\u1ec1 \u0111\u1eb7t ra \u1edf \u0111\u00e2y \u0111\u00e3 s\u00e2u s\u1eafc h\u01a1n.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">V\u00e0o n\u0103m 1831, nh\u00e0 to\u00e1n h\u1ecdc Ph\u00e1p 20 tu\u1ed5i l\u00e0 Galois \u0111\u00e3 \u0111\u01b0a ra m\u1ed9t h\u00ecnh th\u1ee9c tr\u1ea3 l\u1eddi v\u1ea5n \u0111\u1ec1 \u0111\u1eb7t ra m\u1ed9t c\u00e1ch s\u1eafc b\u00e9n v\u00e0 nhanh ch\u00f3ng.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Galois \u0111\u00e3 x\u00e2y d\u1ef1ng n\u00ean l\u00ed thuy\u1ebft nh\u00f3m Galois l\u00e0 c\u01a1 s\u1edf cho \u0111\u1ea1i s\u1ed1 h\u1ecdc hi\u1ec7n \u0111\u1ea1i. D\u1ef1a v\u00e0o l\u00ed thuy\u1ebft nh\u00f3m Galois, Galois \u0111\u00e3 \u0111\u01b0a ra \u0111i\u1ec1u ki\u1ec7n \u0111\u1ec3 m\u1ed9t ph\u01b0\u01a1ng tr\u00ecnh \u0111\u1ea1i s\u1ed1 b\u1eadc cao c\u00f3 th\u1ec3 gi\u1ea3i \u0111\u01b0\u1ee3c b\u1eb1ng c\u0103n th\u1ee9c \u0111\u00f3 l\u00e0 \u201cph\u00e1n \u0111o\u00e1n Galois\u201d. T\u1eeb \u201cph\u00e1n \u0111o\u00e1n Galois\u201d c\u0169ng \u0111i \u0111\u1ebfn k\u1ebft lu\u1eadn l\u00e0 c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh t\u1ed5ng qu\u00e1t b\u1eadc l\u1edbn h\u01a1n ho\u1eb7c b\u1eb1ng 5 (b\u1eadc n \u2265 5) kh\u00f4ng gi\u1ea3i \u0111\u01b0\u1ee3c b\u1eb1ng c\u0103n th\u1ee9c. T\u1eeb \u0111\u00f3 c\u00f3 th\u1ec3 th\u1ea5y r\u1eb1ng \u0111\u1ecbnh l\u00ed Abel ch\u1ec9 l\u00e0 m\u1ed9t h\u1ec7 qu\u1ea3 c\u1ee7a l\u00ed thuy\u1ebft Galois.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">T\u1eeb kho\u00e1: Ph\u00e1n \u0111o\u00e1n Galois v\u00e0 l\u00ed thuy\u1ebft nh\u00f3m.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n\n\n\n","protected":false},"excerpt":{"rendered":"<p>Nhi\u1ec1u ng\u01b0\u1eddi th\u00edch d\u00f9ng c\u00f4ng th\u1ee9c khi gi\u1ea3i c\u00e1c ph\u01b0\u01a1ng tr\u00ecnh v\u00ec ch\u1ec9 c\u1ea7n theo c\u00e1c qu\u00e1 tr\u00ecnh v\u00e0 quy ph\u1ea1m kh\u00f4ng c\u1ea7n ph\u1ea3i t\u1ed1n nhi\u1ec1u suy ngh\u0129. V\u00ed nh\u01b0 gi\u1ea3i ph\u01b0\u01a1ng tr\u00ecnh b\u1eadc hai ax2 + bx + c = 0 (trong \u0111\u00f3 a \u2260 0) ta ch\u1ec9 c\u1ea7n d\u00f9ng c\u00f4ng th\u1ee9c: ta c\u00f3 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":"","_disable_autopaging":false},"categories":[15,9],"tags":[],"class_list":["post-7531","post","type-post","status-publish","format-standard","hentry","category-hoi-dap-khoa-hoc","category-toan-hoc"],"_links":{"self":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7531","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/comments?post=7531"}],"version-history":[{"count":0,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/posts\/7531\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/media?parent=7531"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/categories?post=7531"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vietlearn.org\/kien-thuc\/wp-json\/wp\/v2\/tags?post=7531"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}