{"id":1005,"date":"2020-10-12T22:41:51","date_gmt":"2020-10-12T22:41:51","guid":{"rendered":"http:\/\/localhost\/?page_id=1005"},"modified":"2020-10-12T22:44:06","modified_gmt":"2020-10-12T22:44:06","slug":"exercicios-11","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/grupos-e-corpos\/exercicios-11\/","title":{"rendered":"Exerc\u00edcios 11"},"content":{"rendered":"

$\\newcommand{\\F}{\\mathbb F}\\newcommand{\\E}{\\mathbb E}$1. Sejam $\\F$, $\\E$, $\\mathbb K$ corpos tais que $\\F\\subseteq\\E\\subseteq\\mathbb K$. Mostre que $\\dim_\\F\\mathbb K=(\\dim_\\F\\E)(\\dim_\\E\\mathbb K)$.<\/p>\n

2. Seja $\\F\\subseteq\\E$ uma extens\u00e3o de corpos e seja $\\alpha\\in\\E$. Mostre que as seguintes s\u00e3o
\nequivalentes:<\/p>\n

    \n
  1. $\\dim_\\F \\F(\\alpha)$ \u00e9 finita;<\/li>\n
  2. $\\alpha$ \u00e9 alg\u00e9brico sobre $\\F$.<\/li>\n<\/ol>\n

    3. Usando exerc\u00edcio 2., mostre que os elementos $\\F$-alg\u00e9bricos de $\\E$ formam um corpo.<\/p>\n

    4. Seja $\\zeta=\\exp(2\\pi\\cdot i\/3)$ (terceira raiz da unidade em $\\mathbb C$). Ache o polin\u00f4mio m\u00ednimo de $\\zeta$.<\/p>\n

    5. Mostre que $\\mathbb Q(\\sqrt 2+\\sqrt 3)=\\mathbb Q(\\sqrt 2,\\sqrt 3)$.<\/p>\n

    6. Seja $\\F$ um corpo e $f(x)\\in\\F[x]$. Mostre que as seguintes afirma\u00e7\u00f5es s\u00e3o equivalentes:<\/p>\n

      \n
    1. existe uma extens\u00e3o $\\E$ de $\\F$ tal que $f(x)$ possui uma raiz m\u00faltipla em $\\E$.<\/li>\n
    2. $\\mbox{mdc}(f(x),f'(x))\\neq 1$.<\/li>\n<\/ol>\n

      7. Seja $\\F$ um corpo de carater\u00edstica zero ou um corpo finito e $f(x)\\in\\F[x]$ um polin\u00f4mio irredut\u00edvel. Mostre que $f(x)$ n\u00e3o possui ra\u00edzes m\u00faltiplas em nenhuma extens\u00e3o $\\E$ de $\\F$.<\/p>\n","protected":false},"excerpt":{"rendered":"

      $\\newcommand{\\F}{\\mathbb F}\\newcommand{\\E}{\\mathbb E}$1. Sejam $\\F$, $\\E$, $\\mathbb K$ corpos tais que $\\F\\subseteq\\E\\subseteq\\mathbb K$. Mostre que $\\dim_\\F\\mathbb K=(\\dim_\\F\\E)(\\dim_\\E\\mathbb K)$. 2. Seja $\\F\\subseteq\\E$ uma extens\u00e3o de corpos e seja $\\alpha\\in\\E$. Mostre que as seguintes s\u00e3o equivalentes: $\\dim_\\F \\F(\\alpha)$ \u00e9 finita; $\\alpha$ \u00e9 alg\u00e9brico sobre $\\F$. 3. Usando exerc\u00edcio 2., mostre que os elementos $\\F$-alg\u00e9bricos de $\\E$ formam … Continue reading Exerc\u00edcios 11<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":684,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1005"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=1005"}],"version-history":[{"count":3,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1005\/revisions"}],"predecessor-version":[{"id":1008,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1005\/revisions\/1008"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/684"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=1005"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}