{"id":32352,"date":"2021-05-09T13:00:30","date_gmt":"2021-05-09T13:00:30","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=32352"},"modified":"2025-03-04T01:39:10","modified_gmt":"2025-03-04T01:39:10","slug":"domaines-dintegrite-et-les-nombres-entiers","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/domaines-dintegrite-et-les-nombres-entiers\/","title":{"rendered":"Domaines d&#8217;Int\u00e9grit\u00e9 et les Nombres Entiers"},"content":{"rendered":"<style>\n\tp, ul, ol{\n\ttext-align: justify;\n\t}\n\th1{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\t}\n\th2{\n\ttext-align:center;\n\ttext-transform: uppercase;\n\tfont-size:24pt;\n\t}\n\th3 { \n\t\ttext-align: center;\n\t\ttext-transform: uppercase;\n\t\tfont-size: 24px !important;\n\t}\n<\/style>\n<h1>Domaines d&#8217;Int\u00e9grit\u00e9 et les Nombres Entiers<\/h1>\n<p style=\"text-align:center;\"><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, on introduit le concept de Domaine d&#8217;Int\u00e9grit\u00e9, on explique son importance dans l&#8217;\u00e9tude de l&#8217;alg\u00e8bre g\u00e9n\u00e9rale et on d\u00e9montre, \u00e0 l&#8217;aide de preuves formelles, certaines de ses propri\u00e9t\u00e9s les plus importantes.<\/em><\/p>\n<p style=\"text-align:center;\"><em><strong>Objectifs d&#8217;Apprentissage :<\/strong><\/em><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :\n<\/p>\n<ol>\n<li><strong>Comprendre<\/strong> l&#8217;objectif de l&#8217;\u00e9tude de l&#8217;alg\u00e8bre g\u00e9n\u00e9rale.<\/li>\n<li><strong>Comprendre<\/strong> le concept de domaine d&#8217;int\u00e9grit\u00e9.<\/li>\n<li><strong>Expliquer<\/strong> les aspects fondamentaux communs aux domaines d&#8217;int\u00e9grit\u00e9 et aux nombres entiers.<\/li>\n<li><strong>D\u00e9montrer<\/strong> \u00e0 l&#8217;aide de preuves formelles les propri\u00e9t\u00e9s fondamentales des domaines d&#8217;int\u00e9grit\u00e9.<\/li>\n<\/ol>\n<p style=\"text-align:center;\">\n\t<strong><u>TABLE DES MATI\u00c8RES<\/u><\/strong><br \/>\n\t<a href=\"#1\">L&#8217;OBJECTIF DE L&#8217;ALG\u00c8BRE G\u00c9N\u00c9RALE ET LES CONNAISSANCES PR\u00c9ALABLES<\/a><br \/>\n\t<a href=\"#2\">DES NOMBRES ENTIERS AUX DOMAINES D&#8217;INT\u00c9GRIT\u00c9<\/a><br \/>\n\t<a href=\"#3\">ASPECTS FONDAMENTAUX COMMUNS AUX DOMAINES D&#8217;INT\u00c9GRIT\u00c9 ET AUX NOMBRES ENTIERS<\/a><br \/>\n\t<a href=\"#4\">PROPRI\u00c9T\u00c9S DES DOMAINES D&#8217;INT\u00c9GRIT\u00c9 ET DES NOMBRES ENTIERS<\/a><br \/>\n\t<a href=\"#5\">EXERCICES<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/hxmc1-eXWxU?si=57GADT52JG4fHFT-\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><br \/>\n<\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>L&#8217;objectif de l&#8217;alg\u00e8bre g\u00e9n\u00e9rale et les connaissances pr\u00e9alables<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=183s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">L&#8217;objectif principal de l&#8217;alg\u00e8bre g\u00e9n\u00e9rale<\/span><\/strong><\/a> est l&#8217;\u00e9tude de toute la vari\u00e9t\u00e9 des syst\u00e8mes math\u00e9matiques possibles. Ici, nous \u00e9tudierons plusieurs de ces syst\u00e8mes, et parmi les plus importants figurent les nombres naturels et entiers. \u00c0 travers ces derniers, nous arriverons aux domaines d&#8217;int\u00e9grit\u00e9.<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{N}= \\{1,2,3,4,\\cdots\\}<\/span><\/span><\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{Z}= \\{0,\\pm 1,\\pm 2,\\pm 3,\\pm 4,\\cdots\\}<\/span><\/span><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Des nombres entiers aux domaines d&#8217;int\u00e9grit\u00e9<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=358s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Nous commencerons notre \u00e9tude avec les nombres entiers,<\/span><\/strong><\/a> et la raison de proc\u00e9der ainsi est qu&#8217;ils poss\u00e8dent le plus grand nombre de similitudes avec la majorit\u00e9 des syst\u00e8mes num\u00e9riques que nous examinerons dans cette \u00e9tude.<\/p>\n<p>Plut\u00f4t que d&#8217;essayer de d\u00e9finir ce que sont les nombres entiers, nous commencerons en supposant que, quelle que soit leur nature, ils satisfont certaines propri\u00e9t\u00e9s. Pour cela, un ensemble d&#8217;axiomes est choisi de mani\u00e8re \u00e0 pouvoir en d\u00e9duire toutes les propri\u00e9t\u00e9s que nous associons intuitivement aux entiers.<\/p>\n<p>Toutes ces notions sont introduites \u00e0 travers les <strong>axiomes de Peano<\/strong> des naturels en \u00e9tablissant les op\u00e9rations de base de l&#8217;arithm\u00e9tique. En suivant cette m\u00e9thode axiomatique et en \u00e9tendant les diff\u00e9rentes op\u00e9rations sur les naturels et les entiers, on obtient de nouveaux ensembles num\u00e9riques, tels que les rationnels, irrationnels, r\u00e9els, complexes, quaternions, octonions, et bien d&#8217;autres encore.<\/p>\n<p>Ensuite, si nous observons les nombres entiers, nous verrons qu&#8217;ils poss\u00e8dent des propri\u00e9t\u00e9s qui se retrouvent dans tous les autres ensembles num\u00e9riques, comme l&#8217;existence d&#8217;un \u00e9l\u00e9ment neutre pour la multiplication, d&#8217;un neutre additif et des lois distributives. En nous r\u00e9f\u00e9rant \u00e0 ces notions, nous pouvons \u00e9tablir un langage permettant de parler simultan\u00e9ment de tous ces ensembles. C&#8217;est dans ce contexte que surgissent des termes comme :<\/p>\n<ul>\n<li>Domaine d&#8217;int\u00e9grit\u00e9<\/li>\n<li>Anneau<\/li>\n<li>Groupe<\/li>\n<li>Espace vectoriel<\/li>\n<\/ul>\n<p>Et bien d&#8217;autres termes de ce genre&#8230; Nous concentrerons nos efforts sur l&#8217;\u00e9tude des <strong>Domaines d&#8217;Int\u00e9grit\u00e9.<\/strong><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Aspects fondamentaux communs aux domaines d&#8217;int\u00e9grit\u00e9 et aux nombres entiers<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=472s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Pour expliquer ce qu&#8217;est un domaine d&#8217;int\u00e9grit\u00e9<\/span><\/strong><\/a>, nous nous appuierons sur les propri\u00e9t\u00e9s que nous comprenons tr\u00e8s bien \u00e0 partir des nombres entiers. Dans ce contexte, nous avons que si <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> sont des nombres entiers, alors les <strong>lois<\/strong> suivantes sont satisfaites :<\/p>\n<ol>\n<li><strong>Commutatives :<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = b + a<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab = ba<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Associatives :<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a+b+c = (a+b)+c<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(ab)c = abc = a(bc)<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<li><strong>Distributives :<\/strong>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+(b+c) = a(b+c) = ab+ac<\/span><\/span><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>En plus de cela, il existe certains \u00e9l\u00e9ments sp\u00e9ciaux connus sous le nom de neutres.<\/p>\n<ul>\n<li><strong>Neutre additif :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = a \\leftrightarrow c=0<\/span><\/span><\/li>\n<li><strong>Neutre multiplicatif :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ac = a \\leftrightarrow c=1<\/span><\/span><\/li>\n<\/ul>\n<p>L&#8217;objet dont le symbole est <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> est le neutre additif, et celui qui correspond au symbole <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> est le neutre multiplicatif.<\/p>\n<p>Les entiers poss\u00e8dent \u00e9galement des inverses additifs. \u00c0 chaque entier correspond un inverse additif qui, lorsqu&#8217;il est additionn\u00e9 avec lui, donne le neutre additif.<\/p>\n<ul>\n<li><strong>Inverse additif :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+ c = 0 \\longleftrightarrow c=-a<\/span><\/span><\/li>\n<\/ul>\n<p>Les inverses additifs sont reconnaissables par le signe \u00ab-\u00bb qui les accompagne.<\/p>\n<p>Enfin, il existe une <strong>loi de simplification<\/strong> qui s&#8217;exprime par la relation :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(c\\neq 0 \\wedge ca = cb) \\longleftrightarrow (a=b)<\/span><\/span><\/p>\n<p>Ces propri\u00e9t\u00e9s que nous avons examin\u00e9es s&#8217;appliquent \u00e0 de nombreux autres ensembles : r\u00e9els, complexes, polyn\u00f4mes, etc. Ainsi, nous appelons <strong>Domaine d&#8217;int\u00e9grit\u00e9<\/strong> tout ensemble qui satisfait ces propri\u00e9t\u00e9s.<\/p>\n<p><span style=\"color: #800000;\"><strong>D\u00c9FINITION :<\/strong><\/span> Un Domaine d&#8217;int\u00e9grit\u00e9 est tout ensemble <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> muni d&#8217;une op\u00e9ration d&#8217;addition et de multiplication telles que<\/p>\n<ul>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow a+b \\in D<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b\\in D \\longrightarrow ab \\in D<\/span><\/span><\/li>\n<\/ul>\n<p>Et en plus, les lois <strong>associatives, commutatives<\/strong> et <strong>distributives<\/strong> sont satisfaites, <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span> contient des <strong>neutres additifs<\/strong> et <strong>multiplicatifs<\/strong> (chacun de ces \u00e9l\u00e9ments est unique), et enfin, la <strong>loi de simplification<\/strong> est valable.<\/p>\n<h4>Exemple de Domaine d&#8217;Int\u00e9grit\u00e9<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=hxmc1-eXWxU&amp;t=749s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consid\u00e9rons l&#8217;ensemble <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A=\\{a+b\\sqrt{3}\\; |\\; a,b\\in \\mathbb{Z}\\}.<\/span><\/span><\/span> <\/strong><\/a> Cet ensemble, muni des op\u00e9rations usuelles d&#8217;addition et de multiplication, est un domaine d&#8217;int\u00e9grit\u00e9 car il satisfait les lois de commutativit\u00e9, d&#8217;associativit\u00e9 et de distribution, poss\u00e8de un neutre additif et multiplicatif, ainsi qu&#8217;un inverse additif.<\/p>\n<ul>\n<li><strong>Neutre additif :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Neutre multiplicatif :<\/strong> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1+0\\sqrt{3}<\/span><\/span><\/li>\n<li><strong>Inverse additif :<\/strong> Tout \u00e9l\u00e9ment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b\\sqrt{3}<\/span><\/span> poss\u00e8de un inverse additif <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-a-b\\sqrt{3}<\/span><\/span><\/li>\n<\/ul>\n<p>Et le plus important de tout, cet ensemble A est ferm\u00e9 pour les op\u00e9rations d&#8217;addition et de multiplication, dans le sens o\u00f9 si l&#8217;on prend <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x,y\\in A<\/span><\/span>, alors on a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x+y\\in A<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xy\\in A.<\/span><\/span> Cela est facile \u00e0 v\u00e9rifier : Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_1 + b_1\\sqrt{3}<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a_2 + b_2\\sqrt{3}<\/span><\/span> sont des \u00e9l\u00e9ments de <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span>, alors on a :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n(a_1 + b_1\\sqrt{3}) + (a_2 + b_2\\sqrt{3}) &amp;=(a_1+a_2) + (b_1 + b_2)\\sqrt{3} \\in A\\\\ \\\\\n\n(a_1 + b_1\\sqrt{3})  (a_2 + b_2\\sqrt{3})  &amp;= a_1a_2 + a_1b_2\\sqrt{3}+b_1a_2\\sqrt{3} + 3b_1b_2 \\\\\n\n&amp;=(a_1a_2 + 3b_1b_2) + (a_1b_2 + b_1a_2)\\sqrt{3} \\in A\n\n\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/y6jXNPhjKv4?si=9SaXhWHN42sC73lZ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Propri\u00e9t\u00e9s des Domaines d&#8217;Int\u00e9grit\u00e9 et des Nombres Entiers<\/h3>\n<h4>Le neutre additif d&#8217;un domaine d&#8217;int\u00e9grit\u00e9 est unique<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=26s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cela peut \u00eatre d\u00e9montr\u00e9 par r\u00e9duction \u00e0 l&#8217;absurde :<\/span><\/strong><\/a> Supposons qu&#8217;il existe deux neutres additifs, notons-les <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0^\\prime<\/span><\/span>. Alors, on a :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 0\\neq 0^\\prime &amp; \\text{; Hypoth\u00e8se}\\\\\n\n(2) &amp; a+0 = a &amp; \\text{; Hypoth\u00e8se : $0$ est neutre additif}\\\\\n\n(3) &amp; b+0^\\prime = b &amp; \\text{; Hypoth\u00e8se : $0^\\prime$ est neutre additif}\\\\\n\n(4) &amp; 0^\\prime + 0 = 0^\\prime &amp; \\text{; En substituant $a=0^\\prime$ dans $(2)$}\\\\\n\n(5) &amp; 0 + 0^\\prime = 0 &amp; \\text{; En substituant $b=0$ dans $(3)$}\\\\\n\n(6) &amp; 0 = 0^\\prime  &amp; \\text{; D&#039;apr\u00e8s $(4,5)$ et la commutativit\u00e9 de l&#039;addition}\\\\\n\n(7) &amp; \\bot &amp;\\text{; D&#039;apr\u00e8s $(1,6)$}\n\n\\end{array}<\/span>\n<p>\u00c0 partir de ce raisonnement, nous concluons donc :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{0 \\neq 0^\\prime, a + 0 = a, b + 0^\\prime = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Ensuite, par r\u00e9duction \u00e0 l&#8217;absurde, on obtient :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{a + 0 = a, b + 0^\\prime = b\\}\\vdash 0 = 0^\\prime.<\/span><\/span><\/p>\n<p>Autrement dit, s&#8217;il y a deux neutres additifs, alors ils sont \u00e9gaux, et par cons\u00e9quent, il est unique.<\/p>\n<h4>Le neutre multiplicatif est \u00e9galement unique<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=305s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La d\u00e9monstration est pratiquement analogue \u00e0 la pr\u00e9c\u00e9dente.<\/span><\/strong><\/a> S&#8217;il en existait deux, notons-les <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1^\\prime<\/span><\/span>, alors on pourrait raisonner comme suit :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; 1\\neq 1^\\prime &amp; \\text{; Hypoth\u00e8se}\\\\\n\n(2) &amp; 1\\cdot a = a &amp; \\text{; Hypoth\u00e8se : $1$ est neutre multiplicatif}\\\\\n\n(3) &amp; 1^\\prime \\cdot b = b &amp; \\text{; Hypoth\u00e8se : $1^\\prime$ est neutre multiplicatif}\\\\\n\n(4) &amp; 1\\cdot 1^\\prime = 1^\\prime &amp; \\text{; En substituant $a=1^\\prime$ dans $(2)$}\\\\\n\n(5) &amp; 1^\\prime \\cdot 1 = 1 &amp; \\text{; En substituant $b=1$ dans $(3)$}\\\\\n\n(6) &amp; 1 = 1^\\prime  &amp; \\text{; D&#039;apr\u00e8s $(4,5)$ et la commutativit\u00e9 de la multiplication}\\\\\n\n(7) &amp; \\bot &amp;\\text{; D&#039;apr\u00e8s $(1,6)$}\n\n\\end{array}<\/span>\n<p>Nous arrivons ainsi \u00e0 la conclusion suivante :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> \\{1 \\neq 1^\\prime, 1a= a, 1b = b\\}\\vdash \\bot.<\/span><\/span><\/p>\n<p>Ensuite, par r\u00e9duction \u00e0 l&#8217;absurde, on obtient :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{1a= a, 1b= b\\}\\vdash 1 = 1^\\prime.<\/span><\/span><\/p>\n<p>Autrement dit, s&#8217;il y a deux neutres multiplicatifs, alors ils sont \u00e9gaux, et par cons\u00e9quent, il est unique.<\/p>\n<h4>La loi de simplification pour les additions est valide<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=461s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">C&#8217;est ce que nous faisons lorsque<\/span><\/strong><\/a> nous \u00e9liminons des termes dans une \u00e9galit\u00e9 :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b = a+c \\longleftrightarrow b = c<\/span><\/span><\/p>\n<p>Il n&#8217;est pas difficile de d\u00e9montrer cette situation, il suffit de suivre le raisonnement suivant :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a+b = a+c &amp; \\text{; Hypoth\u00e8se} \\\\\n\n(2) &amp; a+b-a = a+c-a &amp; \\text{; De$(1)$, en ajoutant $-a$ des deux c\u00f4t\u00e9s} \\\\\n\n(3) &amp; (a-a)+b = (a-a)+c &amp; \\text{; De$(2)$, commutativit\u00e9 et associativit\u00e9} \\\\\n\n(4) &amp; 0+b = 0+c &amp; \\text{; De$(3)$ et Inverse Additif} \\\\\n\n(5) &amp; b = c &amp; \\text{; De$(4)$ et Neutre Additif} \\\\\n\n\\end{array}<\/span>\n<p>Comme ce raisonnement peut \u00eatre effectu\u00e9 dans les deux sens en appliquant les m\u00eames \u00e9tapes, on obtient :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a+b=a+c \\dashv \\vdash b=c<\/span><\/span><\/p>\n<p>Ce qui est \u00e9quivalent \u00e0 dire que<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vdash a+b=a+c \\longleftrightarrow b=c<\/span><\/span><\/p>\n<h4>Le neutre additif est aussi un \u00e9l\u00e9ment absorbant pour la multiplication<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=632s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Cela signifie simplement<\/span><\/strong><\/a> que, pour tout <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> dans le domaine d&#8217;int\u00e9grit\u00e9, on a :<\/p>\n<p style=\"text-align:center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 0 = 0<\/span><\/span><\/p>\n<p>Ceci est \u00e9galement facile \u00e0 d\u00e9montrer, en suivant le raisonnement suivant :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+0) &amp; \\text{; Loi distributive} \\\\\n\n(2) &amp; a\\cdot a + a\\cdot 0 = a\\cdot  (a+a-a) &amp; \\text{; De$(1)$ et Inverse Additif} \\\\\n\n(3) &amp; a\\cdot a + a\\cdot 0 = a\\cdot a + a\\cdot a - a\\cdot a &amp; \\text{; De$(2)$ et Distributivit\u00e9} \\\\\n\n(4) &amp;  a\\cdot 0 =  a\\cdot a - a\\cdot a &amp; \\text{; De$(3)$ et Simplification des additions} \\\\\n\n(5) &amp;  a\\cdot 0 =  0 &amp; \\text{; De$(4)$ et Inverse Additif} \\\\\n\n\\end{array}<\/span>\n<h4><strong>Loi des signes :<\/strong><\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=736s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Le produit de quantit\u00e9s de m\u00eame signe<\/span><\/strong><\/a> est toujours positif ; le produit de quantit\u00e9s de signes oppos\u00e9s est toujours n\u00e9gatif. La d\u00e9monstration de cette propri\u00e9t\u00e9 est \u00e9galement simple :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot b = a\\cdot b + 0 &amp; \\text{; Neutre Additif}\\\\\n\n(2) &amp; a\\cdot b = a\\cdot b + (a)\\cdot(-b) - (a)\\cdot(-b) &amp; \\text{; De$(1)$ et Inverse Additif}\\\\\n\n(3) &amp; a\\cdot b = a\\cdot (b -b) - (a)\\cdot(-b) &amp; \\text{; De$(2)$ et Inverse Additif}\\\\\n\n(4) &amp; a\\cdot b = a\\cdot 0 + (-a)\\cdot(-b) &amp; \\text{; De$(3)$ et Inverse Additif}\\\\\n\n(5) &amp; a\\cdot b = (-a)\\cdot(-b) &amp; \\text{; De$(4)$ et \u00c9l\u00e9ment Absorbant Multiplicatif}\\\\\n\n\\end{array}<\/span>\n<p>Par cons\u00e9quent : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> ab = (-a)(-b)<\/span><\/span><\/p>\n<p>Pour les signes oppos\u00e9s, le raisonnement est similaire :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; a\\cdot(-b) = a \\cdot (-b) + 0 &amp; \\text{; Neutre Additif} \\\\\n\n(2) &amp; a\\cdot(-b) = a \\cdot (-b) + a \\cdot b  -  a \\cdot b  &amp; \\text{; De$(1)$ et Inverse Additif} \\\\\n\n(3) &amp; a\\cdot(-b) = a \\cdot (b-b)  -  a \\cdot b  &amp; \\text{; De$(2)$ et Distributivit\u00e9} \\\\\n\n(4) &amp; a\\cdot(-b) = a \\cdot 0  -  a \\cdot b  &amp; \\text{; De$(3)$ et Inverse Additif} \\\\\n\n(5) &amp; a\\cdot(-b) = - a \\cdot b  &amp; \\text{; De$(4)$ et \u00c9l\u00e9ment Absorbant Multiplicatif} \\\\\n\n\\end{array}<\/span>\n<p>Par cons\u00e9quent : <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b) = -a(b)<\/span><\/span><\/p>\n<h4>Si le produit de deux nombres est nul, alors au moins l\u2019un d\u2019eux est nul<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=y6jXNPhjKv4&amp;t=875s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Une autre propri\u00e9t\u00e9 souvent utilis\u00e9e<\/span><\/strong><\/a> est la suivante :<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">ab=0 \\leftrightarrow (a=0 \\vee b=0)<\/span><\/span><\/p>\n<p>Sa d\u00e9monstration est \u00e9galement simple :<\/p>\n<p style=\"text-align:center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n(1) &amp; \\{a=0\\} \\models a\\cdot b = 0  &amp; \\textbf{; \u00c9l\u00e9ment Absorbant Multiplicatif} \\\\\n\n(2) &amp; \\models a=0 \\rightarrow a\\cdot b = 0  &amp;\\text{; TD$(1)$} \\\\\n\n(3) &amp; \\models \\neg (a\\cdot b = 0 ) \\rightarrow \\neg(a=0) &amp;\\text{; CPI$(2)$} \\\\\n\n(4) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(a=0) &amp;\\text{; RTD$(3)$} \\\\\n\n(5) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models   \\neg(b=0) &amp;\\text{; Analogue$(4)$} \\\\\n\n(6) &amp; \\{\\neg (a\\cdot b = 0 ) \\}\\models  \\neg(a=0) \\wedge \\neg(b=0) &amp;\\text{; $\\wedge$-int$(4,5)$} \\\\\n\n(7) &amp; \\models (\\neg (a\\cdot b = 0 )) \\rightarrow \\neg(a=0) \\wedge \\neg(b=0)  &amp;\\text{; TD(6)} \\\\\n\n(8) &amp; \\models \\neg(\\neg(a=0) \\wedge \\neg(b=0) ) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; CPI(7)} \\\\\n\n(9) &amp; \\models (a=0 \\vee b=0) \\rightarrow   (a\\cdot b = 0 ) &amp;\\text{; DM(8)} \\\\\n\n(10)&amp; \\{a\\neq 0 , a\\cdot b=0\\} \\models b=0 &amp; \\textbf{; \u00c9l\u00e9ment Absorbant Multiplicatif}\\\\\n\n(11)&amp; \\{a\\cdot b=0\\} \\models a\\neq 0 \\rightarrow  b=0 &amp; \\text{; TD(10)}\\\\\n\n(12)&amp; \\{a\\cdot b=0\\} \\models \\neg(a\\neq 0) \\vee  b=0 &amp; \\text{; $\\rightarrow$-Def(11)}\\\\\n\n(13)&amp; \\{a\\cdot b=0\\} \\models a=0 \\vee  b=0 &amp; \\text{; DN(12)}\\\\\n\n(14)&amp; \\models (a\\cdot b=0) \\rightarrow (a=0 \\vee  b=0) &amp; \\text{; TD(13)}\\\\\n\n(15)&amp; \\models (a\\cdot b=0) \\leftrightarrow (a=0 \\vee  b=0) &amp; \\text{; De(9,14)}\n\n\\end{array}<\/span>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/KZQ3PXeMlKk?si=O_Hek5KFG853Q6qT\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe><\/center><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercices<\/h2>\n<p>Soient <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">c<\/span> des \u00e9l\u00e9ments quelconques d\u2019un domaine d\u2019int\u00e9grit\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">D<\/span>. Montrez que les propri\u00e9t\u00e9s suivantes sont satisfaites :<\/p>\n<ol>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(-a)=(-1)a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=306s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(a+b)=(-a) + (-b)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=827s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(-b)=-(ab)<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1213s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">-(-a)=a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=1628s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a(b-c) = ab - ac<\/span><\/span> <strong>[PROPOS\u00c9]<\/strong><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(a-b)+(b-c) = a-c<\/span><\/span> <strong>[PROPOS\u00c9]<\/strong><\/li>\n<li>Pour tous les <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\in D<\/span><\/span>, il existe un unique <span class=\"katex-eq\" data-katex-display=\"false\">1<\/span> tel que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\cdot 1 = a<\/span><\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=KZQ3PXeMlKk&amp;t=2029s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong><\/a><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">xx = x \\leftrightarrow (x=1 \\vee x=0)<\/span><\/span> <strong>[PROPOS\u00c9]<\/strong><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Domaines d&#8217;Int\u00e9grit\u00e9 et les Nombres Entiers R\u00e9sum\u00e9 : Dans ce cours, on introduit le concept de Domaine d&#8217;Int\u00e9grit\u00e9, on explique son importance dans l&#8217;\u00e9tude de l&#8217;alg\u00e8bre g\u00e9n\u00e9rale et on d\u00e9montre, \u00e0 l&#8217;aide de preuves formelles, certaines de ses propri\u00e9t\u00e9s les plus importantes. Objectifs d&#8217;Apprentissage : \u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":32317,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":9,"footnotes":""},"categories":[1043,569],"tags":[],"class_list":["post-32352","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebre-generale","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Domaines d&#039;Int\u00e9grit\u00e9 et les Nombres Entiers - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Explorez les Domaines d&#039;Int\u00e9grit\u00e9 en alg\u00e8bre, leurs propri\u00e9t\u00e9s et leur relation avec les nombres entiers \u00e0 travers des d\u00e9monstrations et des exercices.\" \/>\n<meta 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