{"id":1069,"date":"2020-10-25T15:07:49","date_gmt":"2020-10-25T18:07:49","guid":{"rendered":"http:\/\/localhost\/?page_id=1069"},"modified":"2020-10-25T15:09:31","modified_gmt":"2020-10-25T18:09:31","slug":"exercicios-13","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/grupos-e-corpos\/exercicios-13\/","title":{"rendered":"Exerc\u00edcios 13"},"content":{"rendered":"

$\\newcommand{\\F}{\\mathbb F}\\newcommand{\\E}{\\mathbb E}\\newcommand{\\Q}{\\mathbb Q}\\newcommand{\\R}{\\mathbb R}\\newcommand{\\C}{\\mathbb C}\\newcommand{\\fix}[1]{\\mbox{Fix}(#1)}\\newcommand{\\gal}[2]{\\mbox{Gal}(#1:#2)}\\newcommand{\\aut}[1]{\\mbox{Aut}(#1)}$1.\u00a0Seja $\\E:\\F$ uma extens\u00e3o, $f(x)\\in\\F[x]$, $\\alpha\\in\\E$ tal que $f(\\alpha)=0$, e $\\varphi\\in\\gal\\E\\F$. Mostre que $f(\\varphi(\\alpha))=0$.<\/p>\n

2.\u00a0Seja $\\E$ um corpo e sejam $\\varphi_1,\\ldots,\\varphi_k\\in\\aut\\E$. Assuma que $\\alpha_1,\\ldots,\\alpha_k\\in\\E$ tais que $\\alpha_1\\varphi_1+\\cdots+\\alpha_k\\varphi_k=0$. Mostre que $\\alpha_1=\\cdots=\\alpha_k=0$.<\/p>\n

3.\u00a0Seja $\\F$ um corpo infinito (por exemplo um corpo de carater\u00edstica zero), $\\E$ uma extens\u00e3o de $\\F$ e sejam $\\alpha,\\beta\\in\\E$ elementos alg\u00e9bricos sobre $\\F$. Mostre que existe um elemento $\\gamma\\in\\E$ tal que $\\F(\\alpha,\\beta)=\\F(\\gamma)$.<\/p>\n

4. Seja $\\E:\\F$ uma extens\u00e3o de corpos finitos. Determine $\\gal\\E\\F$.<\/p>\n

5. Determine o grupo de Galois dos corpos de decomposi\u00e7\u00e3o dos seguintes polin\u00f4mios sobre $\\Q$:<\/p>\n

    \n
  1. $x^3-1$;<\/li>\n
  2. $x^3+1$;<\/li>\n
  3. $x^4+1$.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    $\\newcommand{\\F}{\\mathbb F}\\newcommand{\\E}{\\mathbb E}\\newcommand{\\Q}{\\mathbb Q}\\newcommand{\\R}{\\mathbb R}\\newcommand{\\C}{\\mathbb C}\\newcommand{\\fix}[1]{\\mbox{Fix}(#1)}\\newcommand{\\gal}[2]{\\mbox{Gal}(#1:#2)}\\newcommand{\\aut}[1]{\\mbox{Aut}(#1)}$1.\u00a0Seja $\\E:\\F$ uma extens\u00e3o, $f(x)\\in\\F[x]$, $\\alpha\\in\\E$ tal que $f(\\alpha)=0$, e $\\varphi\\in\\gal\\E\\F$. Mostre que $f(\\varphi(\\alpha))=0$. 2.\u00a0Seja $\\E$ um corpo e sejam $\\varphi_1,\\ldots,\\varphi_k\\in\\aut\\E$. Assuma que $\\alpha_1,\\ldots,\\alpha_k\\in\\E$ tais que $\\alpha_1\\varphi_1+\\cdots+\\alpha_k\\varphi_k=0$. Mostre que $\\alpha_1=\\cdots=\\alpha_k=0$. 3.\u00a0Seja $\\F$ um corpo infinito (por exemplo um corpo de carater\u00edstica zero), $\\E$ uma extens\u00e3o de $\\F$ e sejam $\\alpha,\\beta\\in\\E$ … Continue reading Exerc\u00edcios 13<\/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\/1069"}],"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=1069"}],"version-history":[{"count":3,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1069\/revisions"}],"predecessor-version":[{"id":1071,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/1069\/revisions\/1071"}],"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=1069"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}