. . . "En alg\u00E8bre abstraite, le degr\u00E9 de transcendance d'une extension de corps L/K est une mesure assez grossi\u00E8re de la \u00AB taille \u00BB de l'extension. Plus pr\u00E9cis\u00E9ment, il est d\u00E9fini comme la cardinalit\u00E9 maximale d'un sous-ensemble alg\u00E9briquement ind\u00E9pendant de L sur K. Un sous-ensemble S de L est une base de transcendance de L/K s'il est alg\u00E9briquement ind\u00E9pendant sur K et si de plus L est une extension alg\u00E9brique du corps K (S) (le corps obtenu en adjoignant les \u00E9l\u00E9ments de S \u00E0 K). On peut montrer que chaque extension de corps a une base de transcendance, et que toutes les bases de transcendance ont la m\u00EAme cardinalit\u00E9 ; cette cardinalit\u00E9 est \u00E9gale au degr\u00E9 de transcendance de l'extension et est not\u00E9e trdegK L ou trdeg(L/K). Si aucun corps K n'est sp\u00E9cifi\u00E9, le degr\u00E9 de transcendance d'un corps L est son degr\u00E9 par rapport au corps premier de m\u00EAme caract\u00E9ristique, c'est-\u00E0-dire le corps de nombres rationnels Q si L est de caract\u00E9ristique 0 et le corps fini Fp si L est de caract\u00E9ristique p. L'extension de corps L/K est purement transcendantale s'il existe un sous-ensemble S de L alg\u00E9briquement ind\u00E9pendant sur K et tel que L = K(S)."@fr . . . . "6943"^^ . . . "\u8D85\u8D8A\u6B21\u6570\uFF08\u3061\u3087\u3046\u3048\u3064\u3058\u3059\u3046\u3001\u82F1: transcendence degree\uFF09\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u4F53\u306E\u62E1\u5927 L/K \u306E\u300C\u5927\u304D\u3055\u300D\u306E\u3042\u308B\u7A2E\u306E\u304B\u306A\u308A\u7C97\u3044\u306F\u304B\u308A\u65B9\u3067\u3042\u308B\u3002\u304D\u3061\u3093\u3068\u8A00\u3048\u3070\u3001K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u306A L \u306E\u90E8\u5206\u96C6\u5408\u306E\u6700\u3082\u5927\u304D\u3044\u6FC3\u5EA6\u3068\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u3002 L \u306E\u90E8\u5206\u96C6\u5408 S \u304C L/K \u306E\u8D85\u8D8A\u57FA\u5E95\uFF08transcendence basis\uFF09\u3067\u3042\u308B\u3068\u306F\u3001S \u304C K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u3067\u3001\u3055\u3089\u306B L \u304C\u4F53 K(S)\uFF08K \u306B S \u306E\u5143\u3092\u6DFB\u52A0\u3057\u3066\u5F97\u3089\u308C\u308B\u4F53\uFF09\u306E\u4EE3\u6570\u62E1\u5927\u3067\u3042\u308B\u3068\u304D\u306B\u3044\u3046\u3002\u3059\u3079\u3066\u306E\u4F53\u62E1\u5927\u306F\u8D85\u8D8A\u57FA\u5E95\u3092\u3082\u3061\u3001\u3059\u3079\u3066\u306E\u8D85\u8D8A\u57FA\u5E95\u306F\u540C\u3058\u6FC3\u5EA6\u3092\u3082\u3064\u3053\u3068\u3092\u8A3C\u660E\u3067\u304D\u308B\u3002\u3053\u306E\u6FC3\u5EA6\u306F\u62E1\u5927\u306E\u8D85\u8D8A\u6B21\u6570\u306B\u7B49\u3057\u304F\u3001trdegK L \u3084 trans. degK L, trdeg(L /K) \u306A\u3069\u3068\u8868\u8A18\u3055\u308C\u308B\u3002 \u4F53 K \u304C\u6307\u5B9A\u3055\u308C\u3066\u3044\u306A\u3044\u5834\u5408\u3001\u4F53 L \u306E\u8D85\u8D8A\u6B21\u6570\u306F\u540C\u3058\u6A19\u6570\u306E\u7D20\u4F53\uFF08\u3064\u307E\u308A L \u306E\u6A19\u6570\u304C 0 \u306A\u3089 Q\u3001L \u306E\u6A19\u6570\u304C\u7D20\u6570 p \u306A\u3089 Fp\uFF09\u4E0A\u306E\u6B21\u6570\u3067\u3042\u308B\u3002 \u4F53\u62E1\u5927 L/K \u306F\u3001K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u3067\u3001L = K(S) \u3067\u3042\u308B\u3088\u3046\u306A\u3001L \u306E\u3042\u308B\u90E8\u5206\u96C6\u5408 S \u304C\u5B58\u5728\u3059\u308B\u3068\u304D\u306B\u3001\u7D14\u8D85\u8D8A\u7684\uFF08purely transcendental\uFF09\u3068\u8A00\u3046\u3002"@ja . . . "287364"^^ . . . . . . . . . . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u4E00\u500B\u57DF\u64F4\u5F35 \u7684\u8D85\u8D8A\u6B21\u6578\u662F \u4E2D\u5728 \u4E0A\u4EE3\u6578\u7368\u7ACB\u5B50\u96C6\u7684\u6975\u5927\u57FA\u6578\u3002"@zh . . "Transcendence degree"@en . "\u0421\u0442\u0435\u043F\u0435\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0438 \u2014 \u043C\u0430\u043A\u0441\u0438\u043C\u0430\u043B\u044C\u043D\u043E\u0435 \u0447\u0438\u0441\u043B\u043E \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u0438 \u043D\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043C\u044B\u0445 \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u043E\u0432 \u0432 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0438 \u043F\u043E\u043B\u044F.\u0421\u0442\u0435\u043F\u0435\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0438 \u0434\u0430\u0451\u0442 \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E\u0441\u0442\u044C \u0438\u0437\u043C\u0435\u0440\u0435\u043D\u0438\u044F \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u044B \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u044F."@ru . . . "Transzendenzbasis"@de . . . . . "\u5728\u62BD\u8C61\u4EE3\u6578\u4E2D\uFF0C\u4E00\u500B\u57DF\u64F4\u5F35 \u7684\u8D85\u8D8A\u6B21\u6578\u662F \u4E2D\u5728 \u4E0A\u4EE3\u6578\u7368\u7ACB\u5B50\u96C6\u7684\u6975\u5927\u57FA\u6578\u3002"@zh . "In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the \"size\" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K. If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., the rational numbers field Q if L is of characteristic 0 and the finite field Fp if L is of characteristic p."@en . . . "\u8D85\u8D8A\u6B21\u6570\uFF08\u3061\u3087\u3046\u3048\u3064\u3058\u3059\u3046\u3001\u82F1: transcendence degree\uFF09\u306F\u62BD\u8C61\u4EE3\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u4F53\u306E\u62E1\u5927 L/K \u306E\u300C\u5927\u304D\u3055\u300D\u306E\u3042\u308B\u7A2E\u306E\u304B\u306A\u308A\u7C97\u3044\u306F\u304B\u308A\u65B9\u3067\u3042\u308B\u3002\u304D\u3061\u3093\u3068\u8A00\u3048\u3070\u3001K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u306A L \u306E\u90E8\u5206\u96C6\u5408\u306E\u6700\u3082\u5927\u304D\u3044\u6FC3\u5EA6\u3068\u3057\u3066\u5B9A\u7FA9\u3055\u308C\u308B\u3002 L \u306E\u90E8\u5206\u96C6\u5408 S \u304C L/K \u306E\u8D85\u8D8A\u57FA\u5E95\uFF08transcendence basis\uFF09\u3067\u3042\u308B\u3068\u306F\u3001S \u304C K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u3067\u3001\u3055\u3089\u306B L \u304C\u4F53 K(S)\uFF08K \u306B S \u306E\u5143\u3092\u6DFB\u52A0\u3057\u3066\u5F97\u3089\u308C\u308B\u4F53\uFF09\u306E\u4EE3\u6570\u62E1\u5927\u3067\u3042\u308B\u3068\u304D\u306B\u3044\u3046\u3002\u3059\u3079\u3066\u306E\u4F53\u62E1\u5927\u306F\u8D85\u8D8A\u57FA\u5E95\u3092\u3082\u3061\u3001\u3059\u3079\u3066\u306E\u8D85\u8D8A\u57FA\u5E95\u306F\u540C\u3058\u6FC3\u5EA6\u3092\u3082\u3064\u3053\u3068\u3092\u8A3C\u660E\u3067\u304D\u308B\u3002\u3053\u306E\u6FC3\u5EA6\u306F\u62E1\u5927\u306E\u8D85\u8D8A\u6B21\u6570\u306B\u7B49\u3057\u304F\u3001trdegK L \u3084 trans. degK L, trdeg(L /K) \u306A\u3069\u3068\u8868\u8A18\u3055\u308C\u308B\u3002 \u4F53 K \u304C\u6307\u5B9A\u3055\u308C\u3066\u3044\u306A\u3044\u5834\u5408\u3001\u4F53 L \u306E\u8D85\u8D8A\u6B21\u6570\u306F\u540C\u3058\u6A19\u6570\u306E\u7D20\u4F53\uFF08\u3064\u307E\u308A L \u306E\u6A19\u6570\u304C 0 \u306A\u3089 Q\u3001L \u306E\u6A19\u6570\u304C\u7D20\u6570 p \u306A\u3089 Fp\uFF09\u4E0A\u306E\u6B21\u6570\u3067\u3042\u308B\u3002 \u4F53\u62E1\u5927 L/K \u306F\u3001K \u4E0A\u4EE3\u6570\u7684\u306B\u72EC\u7ACB\u3067\u3001L = K(S) \u3067\u3042\u308B\u3088\u3046\u306A\u3001L \u306E\u3042\u308B\u90E8\u5206\u96C6\u5408 S \u304C\u5B58\u5728\u3059\u308B\u3068\u304D\u306B\u3001\u7D14\u8D85\u8D8A\u7684\uFF08purely transcendental\uFF09\u3068\u8A00\u3046\u3002"@ja . . . . . . "\u8D85\u8D8A\u6B21\u6578"@zh . . "Transzendenzbasis ist ein algebraischer Begriff aus der Theorie der K\u00F6rpererweiterungen, der in Analogie zum Begriff der Vektorraumbasis der linearen Algebra gesehen werden kann. Die M\u00E4chtigkeit einer solchen Transzendenzbasis, der sogenannte Transzendenzgrad, stellt ein Ma\u00DF f\u00FCr die Gr\u00F6\u00DFe einer transzendenten K\u00F6rpererweiterung dar."@de . "Em \u00E1lgebra abstrata, o grau de transcend\u00EAncia de uma extens\u00E3o de corpo L / K \u00E9 uma certa medida bastante grosseira do \"tamanho\" da extens\u00E3o. Especificamente, ele define a maior cardinalidade de um subconjunto algebricamente independente de L sobre K. \u00C9 poss\u00EDvel mostrar que esta defini\u00E7\u00E3o faz sentido, ou seja, que existe um conjunto maximal de elementos algebricamente independentes (o que requer o axioma da escolha), e que dois destes conjuntos tem a mesma cardinalidade."@pt . . . . . "\u0421\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u043F\u043E\u043B\u044F \u0446\u0435 \u043D\u0430\u0439\u0431\u0456\u043B\u044C\u0448\u0430 \u043F\u043E\u0442\u0443\u0436\u043D\u0456\u0441\u0442\u044C \u043F\u0456\u0434\u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u043F\u043E\u043B\u044F , \u0449\u043E \u0454 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u043E\u044E \u0449\u043E\u0434\u043E \u043F\u043E\u043B\u044F . \u0420\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u0454 \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u0438\u043C \u0442\u043E\u0434\u0456 \u0439 \u043B\u0438\u0448\u0435 \u0442\u043E\u0434\u0456, \u043A\u043E\u043B\u0438 \u043F\u043E\u043B\u0435 \u043C\u0456\u0441\u0442\u0438\u0442\u044C \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438, \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u0456 \u043D\u0430\u0434 , \u0442\u043E\u0431\u0442\u043E \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438, \u0449\u043E \u043D\u0435 \u0454 \u043A\u043E\u0440\u0435\u043D\u0435\u043C \u043D\u0456\u044F\u043A\u043E\u0433\u043E \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E\u0433\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438 \u0437 . \u0412\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u0454 \u0430\u043B\u0433\u0435\u0431\u0440\u0438\u0447\u043D\u0438\u043C \u0442\u043E\u0434\u0456 \u0439 \u043B\u0438\u0448\u0435 \u0442\u043E\u0434\u0456 \u043A\u043E\u043B\u0438 \u0439\u043E\u0433\u043E \u0441\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u0456\u0432\u043D\u0438\u0439 \u043D\u0443\u043B\u044E. \u042F\u043A\u0449\u043E \u2014 \u043C\u0430\u043A\u0441\u0438\u043C\u0430\u043B\u044C\u043D\u0430 \u043C\u043D\u043E\u0436\u0438\u043D\u0430, \u0432\u0441\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u044F\u043A\u043E\u0457 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u0456, \u0442\u043E \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0431\u0430\u0437\u0438\u0441\u043E\u043C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u043F\u043E\u043B\u044F \u043D\u0430\u0434 .\u0423\u0441\u0456 \u0431\u0430\u0437\u0438\u0441\u0438 \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u043C\u0430\u044E\u0442\u044C \u043E\u0434\u043D\u0430\u043A\u043E\u0432\u0443 \u043F\u043E\u0442\u0443\u0436\u043D\u0456\u0441\u0442\u044C, \u0449\u043E \u0440\u0456\u0432\u043D\u0430 \u0441\u0442\u0435\u043F\u0435\u043D\u044E \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F. \u0414\u043B\u044F \u043F\u043E\u043B\u0456\u0432 \u0441\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u0456\u0432\u043D\u0438\u0439 \u0441\u0443\u043C\u0456 \u0441\u0442\u0435\u043F\u0435\u043D\u0456\u0432 \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0442\u0430 . \u042F\u043A\u0449\u043E \u0432\u0441\u0456 \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438 \u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u0456, \u0442\u043E \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u0434\u043E \u043D\u0430\u0437\u0438\u0432\u0430\u0454\u0442\u044C\u0441\u044F \u0447\u0438\u0441\u0442\u043E \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u0438\u043C. \u0412 \u0446\u044C\u043E\u043C\u0443 \u0432\u0438\u043F\u0430\u0434\u043A\u0443 \u043F\u043E\u043B\u0435 \u0456\u0437\u043E\u043C\u043E\u0440\u0444\u043D\u0435 \u043F\u043E\u043B\u044E \u0440\u0430\u0446\u0456\u043E\u043D\u0430\u043B\u044C\u043D\u0438\u0445 \u0444\u0443\u043D\u043A\u0446\u0456\u0439 \u0432\u0456\u0434 \u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u0437\u043C\u0456\u043D\u043D\u0438\u0445 \u043D\u0430\u0434 ."@uk . . . . . "Transzendenzbasis ist ein algebraischer Begriff aus der Theorie der K\u00F6rpererweiterungen, der in Analogie zum Begriff der Vektorraumbasis der linearen Algebra gesehen werden kann. Die M\u00E4chtigkeit einer solchen Transzendenzbasis, der sogenannte Transzendenzgrad, stellt ein Ma\u00DF f\u00FCr die Gr\u00F6\u00DFe einer transzendenten K\u00F6rpererweiterung dar."@de . "\u0421\u0442\u0435\u043F\u0435\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0438"@ru . . . . "\u0421\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u043F\u043E\u043B\u044F \u0446\u0435 \u043D\u0430\u0439\u0431\u0456\u043B\u044C\u0448\u0430 \u043F\u043E\u0442\u0443\u0436\u043D\u0456\u0441\u0442\u044C \u043F\u0456\u0434\u043C\u043D\u043E\u0436\u0438\u043D\u0438 \u043F\u043E\u043B\u044F , \u0449\u043E \u0454 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E \u043D\u0435\u0437\u0430\u043B\u0435\u0436\u043D\u043E\u044E \u0449\u043E\u0434\u043E \u043F\u043E\u043B\u044F . \u0420\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u0454 \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u0438\u043C \u0442\u043E\u0434\u0456 \u0439 \u043B\u0438\u0448\u0435 \u0442\u043E\u0434\u0456, \u043A\u043E\u043B\u0438 \u043F\u043E\u043B\u0435 \u043C\u0456\u0441\u0442\u0438\u0442\u044C \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438, \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u0456 \u043D\u0430\u0434 , \u0442\u043E\u0431\u0442\u043E \u0435\u043B\u0435\u043C\u0435\u043D\u0442\u0438, \u0449\u043E \u043D\u0435 \u0454 \u043A\u043E\u0440\u0435\u043D\u0435\u043C \u043D\u0456\u044F\u043A\u043E\u0433\u043E \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u043E\u0433\u043E \u0440\u0456\u0432\u043D\u044F\u043D\u043D\u044F \u0437 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0430\u043C\u0438 \u0437 . \u0412\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u0440\u043E\u0437\u0448\u0438\u0440\u0435\u043D\u043D\u044F \u0454 \u0430\u043B\u0433\u0435\u0431\u0440\u0438\u0447\u043D\u0438\u043C \u0442\u043E\u0434\u0456 \u0439 \u043B\u0438\u0448\u0435 \u0442\u043E\u0434\u0456 \u043A\u043E\u043B\u0438 \u0439\u043E\u0433\u043E \u0441\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456 \u0440\u0456\u0432\u043D\u0438\u0439 \u043D\u0443\u043B\u044E."@uk . . "Em \u00E1lgebra abstrata, o grau de transcend\u00EAncia de uma extens\u00E3o de corpo L / K \u00E9 uma certa medida bastante grosseira do \"tamanho\" da extens\u00E3o. Especificamente, ele define a maior cardinalidade de um subconjunto algebricamente independente de L sobre K. \u00C9 poss\u00EDvel mostrar que esta defini\u00E7\u00E3o faz sentido, ou seja, que existe um conjunto maximal de elementos algebricamente independentes (o que requer o axioma da escolha), e que dois destes conjuntos tem a mesma cardinalidade. Um subconjunto S de L \u00E9 uma base de transcend\u00EAncia de L/K se \u00E9 algebricamente independente em K e se al\u00E9m disso L \u00E9 uma extens\u00E3o alg\u00E9brica do corpo K(S) (o corpo obtido pela jun\u00E7\u00E3o dos elementos de S a K). Pode-se mostrar que cada extens\u00E3o de corpo tem uma base de transcend\u00EAncia, e que todas bases de transcend\u00EAncia tem a mesma cardinalidade; esta cardinalidade \u00E9 igual ao grau de transcend\u00EAncia da extens\u00E3o e \u00E9 notada trdegK L ou trdeg(L/K) (trdeg do ingl\u00EAs transcendence degree). Se nenhum campo K \u00E9 especificado, o grau de transcend\u00EAncia de um corpo L \u00E9 seu grau relativo ao corpo primo de mesma caracter\u00EDstica, i.e., Q se L \u00E9 de caracter\u00EDstica 0 e Fp se L \u00E9 de caracter\u00EDstica p. A extens\u00E3o de corpo L/K \u00E9 puramente transcendental se existe um subconjunto S de L que \u00E9 algebricamente independente em K e tal que L = K(S)."@pt . . . . . . . "Grau de transcend\u00EAncia"@pt . . . "\u8D85\u8D8A\u6B21\u6570"@ja . . . . . . . . . . . "\u0421\u0442\u0435\u043F\u0456\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0456"@uk . . . . . . . "In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the \"size\" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of L over K. A subset S of L is a transcendence basis of L / K if it is algebraically independent over K and if furthermore L is an algebraic extension of the field K(S) (the field obtained by adjoining the elements of S to K). One can show that every field extension has a transcendence basis, and that all transcendence bases have the same cardinality; this cardinality is equal to the transcendence degree of the extension and is denoted trdegK L or trdeg(L / K). If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., the rational numbers field Q if L is of characteristic 0 and the finite field Fp if L is of characteristic p. The field extension L / K is purely transcendental if there is a subset S of L that is algebraically independent over K and such that L = K(S)."@en . "Degr\u00E9 de transcendence"@fr . . . "1093605733"^^ . . "En alg\u00E8bre abstraite, le degr\u00E9 de transcendance d'une extension de corps L/K est une mesure assez grossi\u00E8re de la \u00AB taille \u00BB de l'extension. Plus pr\u00E9cis\u00E9ment, il est d\u00E9fini comme la cardinalit\u00E9 maximale d'un sous-ensemble alg\u00E9briquement ind\u00E9pendant de L sur K. Si aucun corps K n'est sp\u00E9cifi\u00E9, le degr\u00E9 de transcendance d'un corps L est son degr\u00E9 par rapport au corps premier de m\u00EAme caract\u00E9ristique, c'est-\u00E0-dire le corps de nombres rationnels Q si L est de caract\u00E9ristique 0 et le corps fini Fp si L est de caract\u00E9ristique p."@fr . . "\u0421\u0442\u0435\u043F\u0435\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0438 \u2014 \u043C\u0430\u043A\u0441\u0438\u043C\u0430\u043B\u044C\u043D\u043E\u0435 \u0447\u0438\u0441\u043B\u043E \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u0438 \u043D\u0435\u0437\u0430\u0432\u0438\u0441\u0438\u043C\u044B\u0445 \u044D\u043B\u0435\u043C\u0435\u043D\u0442\u043E\u0432 \u0432 \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u0438 \u043F\u043E\u043B\u044F.\u0421\u0442\u0435\u043F\u0435\u043D\u044C \u0442\u0440\u0430\u043D\u0441\u0446\u0435\u043D\u0434\u0435\u043D\u0442\u043D\u043E\u0441\u0442\u0438 \u0434\u0430\u0451\u0442 \u0432\u043E\u0437\u043C\u043E\u0436\u043D\u043E\u0441\u0442\u044C \u0438\u0437\u043C\u0435\u0440\u0435\u043D\u0438\u044F \u0432\u0435\u043B\u0438\u0447\u0438\u043D\u044B \u0440\u0430\u0441\u0448\u0438\u0440\u0435\u043D\u0438\u044F."@ru . . .