Understanding of a proof












1












$begingroup$


In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.



enter image description here










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.



    enter image description here










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.



      enter image description here










      share|cite|improve this question









      $endgroup$




      In (1) and (3), $ M' subseteq L'$ and $H subseteq H''$ are proved respectively. But I didn't understand how these imply $ M' leq L'$ and $H leq H''$ again respectively.



      enter image description here







      galois-theory extension-field






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 30 '18 at 0:20









      Leyla AlkanLeyla Alkan

      1,5751724




      1,5751724






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.



          Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
            $endgroup$
            – Leyla Alkan
            Dec 30 '18 at 0:45











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056395%2funderstanding-of-a-proof%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.



          Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
            $endgroup$
            – Leyla Alkan
            Dec 30 '18 at 0:45
















          1












          $begingroup$

          I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.



          Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
            $endgroup$
            – Leyla Alkan
            Dec 30 '18 at 0:45














          1












          1








          1





          $begingroup$

          I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.



          Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.






          share|cite|improve this answer









          $endgroup$



          I'm assuming you're reading [this PDF}(http://feyzioglu.boun.edu.tr/book/chapter5/ch5(54).pdf). Thus, Lemma 54.8 implies that $M', L'$ are indeed subgroups of $G$. Thus, if $M' subseteq L'$, then $M'$ is a group contained within the group $L'$. In other words, $M'$ is a subgroup of $L'$, and thus $M' leq L'$.



          Also according to Lemma 54.8, $H'$ is an intermediate field of $E/K$ and thus $H''$ is a subgroup of $G$. Also, by definition, $H$ is a subgroup of $G$. Therefore, $H subseteq H''$ implies that $H$ is a group contained within the group $H''$. Similarly to above, in other words, this means that $H$ is a subgroup of $H''$ and thus $H leq H''$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 30 '18 at 0:35









          Noble MushtakNoble Mushtak

          15.3k1835




          15.3k1835












          • $begingroup$
            I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
            $endgroup$
            – Leyla Alkan
            Dec 30 '18 at 0:45


















          • $begingroup$
            I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
            $endgroup$
            – Leyla Alkan
            Dec 30 '18 at 0:45
















          $begingroup$
          I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
          $endgroup$
          – Leyla Alkan
          Dec 30 '18 at 0:45




          $begingroup$
          I totally dismissed the use of Lemma 54.8 there. I really appreciated your help, thanks a lot!
          $endgroup$
          – Leyla Alkan
          Dec 30 '18 at 0:45


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056395%2funderstanding-of-a-proof%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Probability when a professor distributes a quiz and homework assignment to a class of n students.

          Aardman Animations

          Are they similar matrix