Let $z = e^ frac 2pi i7 $ and let $p = z+z^2+z^4 $.Then which of the following options are correct?cube root of 2 not in Q(primitive root)For which of the following fields $mathbb F$ the polynomial $x^3-312312x+123123$ is irreducible in $mathbb F[x]$?Perfect field of characteristic $p>0$ which is not an algebraic extension of the prime fieldLet $F$ be a field of 8 elements and $A$= $xin F$. Then the number of elements in A isWhich of these statements about the field extension $mathbbR/mathbbQ$ are true?Let $(F,+,cdot)$ is the finite field with $9$ elements. Then which of the following are true?Which of the following field properties are correct?Are the following options correct in case of a field?Are the extensions $mathbbQ(sqrt2,sqrt3)$ and $mathbbQ(sqrt[3]5)$ normal over $mathbbQ$A problem from Neukirch's algebraic number theory book.

School performs periodic password audits. Is my password compromised?

What's the 'present simple' form of the word "нашла́" in 3rd person singular female?

Can one live in the U.S. and not use a credit card?

I reported the illegal activity of my boss to his boss. My boss found out. Now I am being punished. What should I do?

What do you call someone who likes to pick fights?

Confusion about Complex Continued Fraction

Has a sovereign Communist government ever run, and conceded loss, on a fair election?

Can the alpha, lambda values of a glmnet object output determine whether ridge or Lasso?

Is this Paypal Github SDK reference really a dangerous site?

After `ssh` without `-X` to a machine, is it possible to change `$DISPLAY` to make it work like `ssh -X`?

Do I really need to have a scientific explanation for my premise?

What is Tony Stark injecting into himself in Iron Man 3?

Why is a very small peak with larger m/z not considered to be the molecular ion?

Does an unused member variable take up memory?

Signed and unsigned numbers

How to resolve: Reviewer #1 says remove section X vs. Reviewer #2 says expand section X

Why do we say ‘pairwise disjoint’, rather than ‘disjoint’?

How do electrons receive energy when a body is heated?

Are all players supposed to be able to see each others' character sheets?

Finitely many repeated replacements

How can I manipulate the output of Information?

Is divide-by-zero a security vulnerability?

Proving a statement about real numbers

Getting the || sign while using Kurier



Let $z = e^ frac 2pi i7 $ and let $p = z+z^2+z^4 $.Then which of the following options are correct?


cube root of 2 not in Q(primitive root)For which of the following fields $mathbb F$ the polynomial $x^3-312312x+123123$ is irreducible in $mathbb F[x]$?Perfect field of characteristic $p>0$ which is not an algebraic extension of the prime fieldLet $F$ be a field of 8 elements and $A$= $xin F$. Then the number of elements in A isWhich of these statements about the field extension $mathbbR/mathbbQ$ are true?Let $(F,+,cdot)$ is the finite field with $9$ elements. Then which of the following are true?Which of the following field properties are correct?Are the following options correct in case of a field?Are the extensions $mathbbQ(sqrt2,sqrt3)$ and $mathbbQ(sqrt[3]5)$ normal over $mathbbQ$A problem from Neukirch's algebraic number theory book.













1












$begingroup$


Let $z= e^ frac 2pi i7 $ and let $p= z+z^2+z^4 $ then



  1. $p$ is in $ mathbb Q $


  2. $p$ is in $ mathbbQ (sqrt D) $ for some $D gt 0$


  3. $p$ is in $ mathbbQ(sqrt D) $ for some $D lt 0$


  4. $p$ is in $i mathbb R $


Option $1$ is clearly false. please give me some hints for other options.



Thanks in advance.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Look up Gauss sums. This local search should give you enough.
    $endgroup$
    – Jyrki Lahtonen
    yesterday















1












$begingroup$


Let $z= e^ frac 2pi i7 $ and let $p= z+z^2+z^4 $ then



  1. $p$ is in $ mathbb Q $


  2. $p$ is in $ mathbbQ (sqrt D) $ for some $D gt 0$


  3. $p$ is in $ mathbbQ(sqrt D) $ for some $D lt 0$


  4. $p$ is in $i mathbb R $


Option $1$ is clearly false. please give me some hints for other options.



Thanks in advance.










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Look up Gauss sums. This local search should give you enough.
    $endgroup$
    – Jyrki Lahtonen
    yesterday













1












1








1


1



$begingroup$


Let $z= e^ frac 2pi i7 $ and let $p= z+z^2+z^4 $ then



  1. $p$ is in $ mathbb Q $


  2. $p$ is in $ mathbbQ (sqrt D) $ for some $D gt 0$


  3. $p$ is in $ mathbbQ(sqrt D) $ for some $D lt 0$


  4. $p$ is in $i mathbb R $


Option $1$ is clearly false. please give me some hints for other options.



Thanks in advance.










share|cite|improve this question











$endgroup$




Let $z= e^ frac 2pi i7 $ and let $p= z+z^2+z^4 $ then



  1. $p$ is in $ mathbb Q $


  2. $p$ is in $ mathbbQ (sqrt D) $ for some $D gt 0$


  3. $p$ is in $ mathbbQ(sqrt D) $ for some $D lt 0$


  4. $p$ is in $i mathbb R $


Option $1$ is clearly false. please give me some hints for other options.



Thanks in advance.







field-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday









SNEHIL SANYAL

618110




618110










asked yesterday









suchanda adhikarisuchanda adhikari

807




807







  • 1




    $begingroup$
    Look up Gauss sums. This local search should give you enough.
    $endgroup$
    – Jyrki Lahtonen
    yesterday












  • 1




    $begingroup$
    Look up Gauss sums. This local search should give you enough.
    $endgroup$
    – Jyrki Lahtonen
    yesterday







1




1




$begingroup$
Look up Gauss sums. This local search should give you enough.
$endgroup$
– Jyrki Lahtonen
yesterday




$begingroup$
Look up Gauss sums. This local search should give you enough.
$endgroup$
– Jyrki Lahtonen
yesterday










2 Answers
2






active

oldest

votes


















4












$begingroup$

Note that
$$p^2=z^2+z^4+z+2z^3+2z^5+2z^6=-p-2+2(1+z+z^2+z^3+z^4+z^5+z^6)$$
and $1+z+cdots+z^6=0$.






share|cite|improve this answer









$endgroup$




















    3












    $begingroup$

    Since $p^ast=z^6+z^5+z^3=z^2p$ and $z^2nepm1$, $p$ is neither real nor imaginary. eliminating $1$, $2$ and $4$. As Lord Shark the Unknown already noted, $p^2+p+2implies p=frac-1pmsqrt-72$ so option $3$ is correct.






    share|cite|improve this answer











    $endgroup$












    • $begingroup$
      why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
      $endgroup$
      – suchanda adhikari
      yesterday






    • 1




      $begingroup$
      @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
      $endgroup$
      – J.G.
      yesterday










    • $begingroup$
      yes but is it possible to determine the sign easily?
      $endgroup$
      – suchanda adhikari
      yesterday






    • 1




      $begingroup$
      @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
      $endgroup$
      – Jyrki Lahtonen
      yesterday











    • $begingroup$
      Thank you sir I will do it.
      $endgroup$
      – suchanda adhikari
      yesterday










    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%2f3141005%2flet-z-e-frac-2-pi-i7-and-let-p-zz2z4-then-which-of-the-fol%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    Note that
    $$p^2=z^2+z^4+z+2z^3+2z^5+2z^6=-p-2+2(1+z+z^2+z^3+z^4+z^5+z^6)$$
    and $1+z+cdots+z^6=0$.






    share|cite|improve this answer









    $endgroup$

















      4












      $begingroup$

      Note that
      $$p^2=z^2+z^4+z+2z^3+2z^5+2z^6=-p-2+2(1+z+z^2+z^3+z^4+z^5+z^6)$$
      and $1+z+cdots+z^6=0$.






      share|cite|improve this answer









      $endgroup$















        4












        4








        4





        $begingroup$

        Note that
        $$p^2=z^2+z^4+z+2z^3+2z^5+2z^6=-p-2+2(1+z+z^2+z^3+z^4+z^5+z^6)$$
        and $1+z+cdots+z^6=0$.






        share|cite|improve this answer









        $endgroup$



        Note that
        $$p^2=z^2+z^4+z+2z^3+2z^5+2z^6=-p-2+2(1+z+z^2+z^3+z^4+z^5+z^6)$$
        and $1+z+cdots+z^6=0$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        Lord Shark the UnknownLord Shark the Unknown

        106k1161133




        106k1161133





















            3












            $begingroup$

            Since $p^ast=z^6+z^5+z^3=z^2p$ and $z^2nepm1$, $p$ is neither real nor imaginary. eliminating $1$, $2$ and $4$. As Lord Shark the Unknown already noted, $p^2+p+2implies p=frac-1pmsqrt-72$ so option $3$ is correct.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
              $endgroup$
              – J.G.
              yesterday










            • $begingroup$
              yes but is it possible to determine the sign easily?
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
              $endgroup$
              – Jyrki Lahtonen
              yesterday











            • $begingroup$
              Thank you sir I will do it.
              $endgroup$
              – suchanda adhikari
              yesterday















            3












            $begingroup$

            Since $p^ast=z^6+z^5+z^3=z^2p$ and $z^2nepm1$, $p$ is neither real nor imaginary. eliminating $1$, $2$ and $4$. As Lord Shark the Unknown already noted, $p^2+p+2implies p=frac-1pmsqrt-72$ so option $3$ is correct.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
              $endgroup$
              – J.G.
              yesterday










            • $begingroup$
              yes but is it possible to determine the sign easily?
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
              $endgroup$
              – Jyrki Lahtonen
              yesterday











            • $begingroup$
              Thank you sir I will do it.
              $endgroup$
              – suchanda adhikari
              yesterday













            3












            3








            3





            $begingroup$

            Since $p^ast=z^6+z^5+z^3=z^2p$ and $z^2nepm1$, $p$ is neither real nor imaginary. eliminating $1$, $2$ and $4$. As Lord Shark the Unknown already noted, $p^2+p+2implies p=frac-1pmsqrt-72$ so option $3$ is correct.






            share|cite|improve this answer











            $endgroup$



            Since $p^ast=z^6+z^5+z^3=z^2p$ and $z^2nepm1$, $p$ is neither real nor imaginary. eliminating $1$, $2$ and $4$. As Lord Shark the Unknown already noted, $p^2+p+2implies p=frac-1pmsqrt-72$ so option $3$ is correct.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited yesterday

























            answered yesterday









            J.G.J.G.

            29.4k22846




            29.4k22846











            • $begingroup$
              why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
              $endgroup$
              – J.G.
              yesterday










            • $begingroup$
              yes but is it possible to determine the sign easily?
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
              $endgroup$
              – Jyrki Lahtonen
              yesterday











            • $begingroup$
              Thank you sir I will do it.
              $endgroup$
              – suchanda adhikari
              yesterday
















            • $begingroup$
              why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
              $endgroup$
              – J.G.
              yesterday










            • $begingroup$
              yes but is it possible to determine the sign easily?
              $endgroup$
              – suchanda adhikari
              yesterday






            • 1




              $begingroup$
              @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
              $endgroup$
              – Jyrki Lahtonen
              yesterday











            • $begingroup$
              Thank you sir I will do it.
              $endgroup$
              – suchanda adhikari
              yesterday















            $begingroup$
            why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
            $endgroup$
            – suchanda adhikari
            yesterday




            $begingroup$
            why $ p= frac -1+sqrt -72 $ @J.G. ,please explain, I can't understand how you delete the possibility that p can be $ frac -1-sqrt -72 $
            $endgroup$
            – suchanda adhikari
            yesterday




            1




            1




            $begingroup$
            @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
            $endgroup$
            – J.G.
            yesterday




            $begingroup$
            @suchandaadhikari Sorry, I meant to write $pm$ there, which I do as of my latest edit. We don't need to determine the sign to work out which options are applicable.
            $endgroup$
            – J.G.
            yesterday












            $begingroup$
            yes but is it possible to determine the sign easily?
            $endgroup$
            – suchanda adhikari
            yesterday




            $begingroup$
            yes but is it possible to determine the sign easily?
            $endgroup$
            – suchanda adhikari
            yesterday




            1




            1




            $begingroup$
            @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
            $endgroup$
            – Jyrki Lahtonen
            yesterday





            $begingroup$
            @suchandaadhikari In general, no! Gauss himself spent a while figuring out the correct sign (for primes much larger than $7$), even though the solution is in many a book now. For $7$th roots of unity it is easy. Draw them on the complexplane. Surely you can figure which side of the real axis the sum is!
            $endgroup$
            – Jyrki Lahtonen
            yesterday













            $begingroup$
            Thank you sir I will do it.
            $endgroup$
            – suchanda adhikari
            yesterday




            $begingroup$
            Thank you sir I will do it.
            $endgroup$
            – suchanda adhikari
            yesterday

















            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%2f3141005%2flet-z-e-frac-2-pi-i7-and-let-p-zz2z4-then-which-of-the-fol%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