{"id":16441,"date":"2025-03-11T14:57:26","date_gmt":"2025-03-11T13:57:26","guid":{"rendered":"https:\/\/cyberschool.univ-rennes.fr\/?page_id=16441"},"modified":"2026-03-12T14:35:52","modified_gmt":"2026-03-12T13:35:52","slug":"theorie-des-nombres","status":"publish","type":"page","link":"https:\/\/cyberschool.univ-rennes.fr\/en\/education\/masters-and-postgraduate-programmes\/masters-in-mathematics-and-applications-it-mathematics-and-cryptography\/study-programme-of-the-masters-mathematics-and-applications-it-mathematics-and-cryptography\/theorie-des-nombres\/","title":{"rendered":"Th\u00e9orie des nombres"},"content":{"rendered":"\n
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<\/span><\/span>Course description<\/h2>\n

Le programme commence par le th\u00e9or\u00e8me de r\u00e9ciprocit\u00e9 quadratique, qu\u2019on pourra traiter en utilisant la th\u00e9orie des extensions de corps vue en ALGB. On fera ensuite la th\u00e9orie de Galois appuy\u00e9e par un certain nombre d\u2019exemples. Apr\u00e8s un chapitre sur les r\u00e9seaux euclidiens et la recherche de vecteurs courts avec le th\u00e9or\u00e8me de Hermite ou Minkowski, on abordera les anneaux d\u2019entiers d\u2019un corps de nombres avec leurs plongements r\u00e9els et complexes, avec comme objectif la preuve du th\u00e9or\u00e8me des unit\u00e9s. On traitera en d\u00e9tail les exemples des extensions quadratiques.<\/p>\n

    \n
  1. R\u00e9ciprocit\u00e9 quadratique\n
      \n
    1. Symboles de Legendre et Jacobi<\/li>\n
    2. Preuve de la r\u00e9ciprocit\u00e9<\/li>\n<\/ol>\n<\/li>\n
    3. Th\u00e9orie de Galois\n
        \n
      1. (extension de rupture et de d\u00e9composition connues). Th\u00e9or\u00e8me de l\u2019\u00e9l\u00e9ment primitif<\/li>\n
      2. Th\u00e9or\u00e8me de correspondance<\/li>\n
      3. R\u00e9solubilit\u00e9 par radicaux<\/li>\n
      4. Exemples : corps finis, extensions quadratiques, extensions cyclotomiques, voire extensions de Kummer<\/li>\n<\/ol>\n<\/li>\n
      5. G\u00e9om\u00e9trie des nombres, r\u00e9seaux\n
          \n
        1. Caract\u00e9risation des r\u00e9seaux comme sous-groupes discrets cocompacts<\/li>\n
        2. Covolumes, lien avec l\u2019indice d\u2019un sous-r\u00e9seau<\/li>\n
        3. Th\u00e9or\u00e8mes de Hermite et\/ou Minkowski<\/li>\n
        4. Th\u00e9or\u00e8mes des 2 et 4 carr\u00e9s<\/li>\n<\/ol>\n<\/li>\n
        5. Entiers alg\u00e9briques\n
            \n
          1. Entiers alg\u00e9briques, anneau des entiers d\u2019un corps de nombres, c\u2019est un Z-module de type fini<\/li>\n
          2. Exemples : entiers de Gauss, d\u2019Eisenstein, Z[p2], entiers quadratiques, leurs unit\u00e9s, caract\u00e8re euclidien<\/li>\n
          3. Les anneaux d\u2019entiers quadratiques<\/li>\n
          4. Plongements r\u00e9els et complexes d\u2019un corps de nombres, et conjugu\u00e9s de Galois<\/li>\n
          5. Les entiers comme un r\u00e9seau<\/li>\n<\/ol>\n<\/li>\n
          6. Trace, norme, th\u00e9or\u00e8me des unit\u00e9s\n
              \n
            1. Trace et norme<\/li>\n
            2. \u00c9nonc\u00e9 du th\u00e9or\u00e8me des unit\u00e9s, exemple des entiers quadratiques<\/li>\n
            3. Preuve du th\u00e9or\u00e8me des unit\u00e9s<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

              En fonction du temps disponible, des compl\u00e9ments possibles, par exemple en TD, sont :<\/p>\n

                \n
              • Discriminant et application \u00e0 la d\u00e9termination algorithmique de OK<\/li>\n
              • Factorisation des id\u00e9aux, groupe de classes<\/li>\n
              • Corps p-adiques.<\/li>\n<\/ul>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n\n
                \n
                \n
                \n

                <\/span><\/span>Skills to acquire<\/h2>\n
                  \n
                • Utilisation des extensions de corps pour r\u00e9soudre des probl\u00e8mes alg\u00e9briques.<\/li>\n
                • Utiliser la g\u00e9om\u00e9trie du plongement euclidien de l\u2019anneau des entiers, ainsi que les notions de trace et de norme pour en d\u00e9duire des propri\u00e9t\u00e9s alg\u00e9briques.<\/li>\n<\/ul>\n\n <\/div>\n <\/div>\n<\/section>\n\n\n\n
                  \n
                  \n
                  \n

                  <\/span><\/span>Biography<\/h2>\n

                  Florian Ivorra<\/strong> est ma\u00eetre de conf\u00e9rences \u00e0 l\u2019Universit\u00e9 de Rennes.<\/p>\n\n <\/div>\n <\/div>\n<\/section>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":8,"featured_media":0,"parent":11605,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"acf":[],"yoast_head":"\nTh\u00e9orie des nombres - CyberSchool<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cyberschool.univ-rennes.fr\/en\/education\/masters-and-postgraduate-programmes\/masters-in-mathematics-and-applications-it-mathematics-and-cryptography\/study-programme-of-the-masters-mathematics-and-applications-it-mathematics-and-cryptography\/theorie-des-nombres\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Th\u00e9orie des nombres - 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