Is $Kcong (mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$? [closed]












0












$begingroup$


Is the following statement true/false?




Let $(mathbb{F},+,cdot)$ be the finite field with $9$ elements. Let $K=(mathbb{F},+)$ denote the underlying additive group; then
$$Kcong (mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$$




My attempt: I thinks this statement is false because $gcd(3,3)= 3 neq 1$



Is my logic correct?



Any hints/solution will be appreciated






abstract-algebra







share|cite|improve this question











$endgroup$



closed as off-topic by user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician Dec 8 '18 at 15:19


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 1




    $begingroup$
    Why would $gcd(3, 3)neq 1$ prove the result?
    $endgroup$
    – Arthur
    Dec 3 '18 at 8:39


















0












$begingroup$


Is the following statement true/false?




Let $(mathbb{F},+,cdot)$ be the finite field with $9$ elements. Let $K=(mathbb{F},+)$ denote the underlying additive group; then
$$Kcong (mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$$




My attempt: I thinks this statement is false because $gcd(3,3)= 3 neq 1$



Is my logic correct?



Any hints/solution will be appreciated






abstract-algebra







share|cite|improve this question











$endgroup$



closed as off-topic by user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician Dec 8 '18 at 15:19


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician

If this question can be reworded to fit the rules in the help center, please edit the question.












  • 1




    $begingroup$
    Why would $gcd(3, 3)neq 1$ prove the result?
    $endgroup$
    – Arthur
    Dec 3 '18 at 8:39
















0












0








0





$begingroup$


Is the following statement true/false?




Let $(mathbb{F},+,cdot)$ be the finite field with $9$ elements. Let $K=(mathbb{F},+)$ denote the underlying additive group; then
$$Kcong (mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$$




My attempt: I thinks this statement is false because $gcd(3,3)= 3 neq 1$



Is my logic correct?



Any hints/solution will be appreciated






abstract-algebra







share|cite|improve this question











$endgroup$




Is the following statement true/false?




Let $(mathbb{F},+,cdot)$ be the finite field with $9$ elements. Let $K=(mathbb{F},+)$ denote the underlying additive group; then
$$Kcong (mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$$




My attempt: I thinks this statement is false because $gcd(3,3)= 3 neq 1$



Is my logic correct?



Any hints/solution will be appreciated






abstract-algebra




abstract-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 3 '18 at 8:38









egreg

181k1485202




181k1485202










asked Dec 3 '18 at 8:31









jasminejasmine

1,701417




1,701417




closed as off-topic by user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician Dec 8 '18 at 15:19


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician

If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician Dec 8 '18 at 15:19


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – user21820, Jyrki Lahtonen, Saad, Brahadeesh, ancientmathematician

If this question can be reworded to fit the rules in the help center, please edit the question.








  • 1




    $begingroup$
    Why would $gcd(3, 3)neq 1$ prove the result?
    $endgroup$
    – Arthur
    Dec 3 '18 at 8:39
















  • 1




    $begingroup$
    Why would $gcd(3, 3)neq 1$ prove the result?
    $endgroup$
    – Arthur
    Dec 3 '18 at 8:39










1




1




$begingroup$
Why would $gcd(3, 3)neq 1$ prove the result?
$endgroup$
– Arthur
Dec 3 '18 at 8:39






$begingroup$
Why would $gcd(3, 3)neq 1$ prove the result?
$endgroup$
– Arthur
Dec 3 '18 at 8:39












2 Answers
2






active

oldest

votes


















2












$begingroup$

The statement is true; $mathbb{F}$ is a vector space over the three-element field and has dimension $2$. The same holds for $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$.



It is false that $mathbb{F}cong(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$ as rings.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:42












  • $begingroup$
    @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
    $endgroup$
    – freakish
    Dec 3 '18 at 8:47












  • $begingroup$
    @freakish that mean $mathbb{F_9}$ is True ??
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:48






  • 1




    $begingroup$
    @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
    $endgroup$
    – egreg
    Dec 3 '18 at 8:53










  • $begingroup$
    @egreg thanks u .
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:56



















1












$begingroup$

It's an abelian group with nine elements, where each element has order $3$. There is only one such group (which I suppose is what you actually need to prove).






share|cite|improve this answer









$endgroup$




















    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    The statement is true; $mathbb{F}$ is a vector space over the three-element field and has dimension $2$. The same holds for $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$.



    It is false that $mathbb{F}cong(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$ as rings.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:42












    • $begingroup$
      @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
      $endgroup$
      – freakish
      Dec 3 '18 at 8:47












    • $begingroup$
      @freakish that mean $mathbb{F_9}$ is True ??
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:48






    • 1




      $begingroup$
      @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
      $endgroup$
      – egreg
      Dec 3 '18 at 8:53










    • $begingroup$
      @egreg thanks u .
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:56
















    2












    $begingroup$

    The statement is true; $mathbb{F}$ is a vector space over the three-element field and has dimension $2$. The same holds for $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$.



    It is false that $mathbb{F}cong(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$ as rings.






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:42












    • $begingroup$
      @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
      $endgroup$
      – freakish
      Dec 3 '18 at 8:47












    • $begingroup$
      @freakish that mean $mathbb{F_9}$ is True ??
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:48






    • 1




      $begingroup$
      @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
      $endgroup$
      – egreg
      Dec 3 '18 at 8:53










    • $begingroup$
      @egreg thanks u .
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:56














    2












    2








    2





    $begingroup$

    The statement is true; $mathbb{F}$ is a vector space over the three-element field and has dimension $2$. The same holds for $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$.



    It is false that $mathbb{F}cong(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$ as rings.






    share|cite|improve this answer









    $endgroup$



    The statement is true; $mathbb{F}$ is a vector space over the three-element field and has dimension $2$. The same holds for $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$.



    It is false that $mathbb{F}cong(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})$ as rings.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 3 '18 at 8:40









    egregegreg

    181k1485202




    181k1485202












    • $begingroup$
      @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:42












    • $begingroup$
      @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
      $endgroup$
      – freakish
      Dec 3 '18 at 8:47












    • $begingroup$
      @freakish that mean $mathbb{F_9}$ is True ??
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:48






    • 1




      $begingroup$
      @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
      $endgroup$
      – egreg
      Dec 3 '18 at 8:53










    • $begingroup$
      @egreg thanks u .
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:56


















    • $begingroup$
      @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:42












    • $begingroup$
      @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
      $endgroup$
      – freakish
      Dec 3 '18 at 8:47












    • $begingroup$
      @freakish that mean $mathbb{F_9}$ is True ??
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:48






    • 1




      $begingroup$
      @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
      $endgroup$
      – egreg
      Dec 3 '18 at 8:53










    • $begingroup$
      @egreg thanks u .
      $endgroup$
      – jasmine
      Dec 3 '18 at 8:56
















    $begingroup$
    @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:42






    $begingroup$
    @@@egreq that mean $mathbb{Z} /3mathbb{Z} timesmathbb{Z} /3mathbb{Z}= mathbb{Z_9} $ or $mathbb{F_9} $ ? which one u r saying
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:42














    $begingroup$
    @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
    $endgroup$
    – freakish
    Dec 3 '18 at 8:47






    $begingroup$
    @jasmine It is not true that $mathbb{Z}/3mathbb{Z}timesmathbb{Z}/3mathbb{Z}simeqmathbb{Z}_9$.
    $endgroup$
    – freakish
    Dec 3 '18 at 8:47














    $begingroup$
    @freakish that mean $mathbb{F_9}$ is True ??
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:48




    $begingroup$
    @freakish that mean $mathbb{F_9}$ is True ??
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:48




    1




    1




    $begingroup$
    @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
    $endgroup$
    – egreg
    Dec 3 '18 at 8:53




    $begingroup$
    @jasmine The statement in the question (isomorphism as abelian group) is true. The statement that $(mathbb{Z}/3mathbb{Z})times(mathbb{Z}/3mathbb{Z})congmathbb{Z}/9mathbb{Z}$ is false.
    $endgroup$
    – egreg
    Dec 3 '18 at 8:53












    $begingroup$
    @egreg thanks u .
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:56




    $begingroup$
    @egreg thanks u .
    $endgroup$
    – jasmine
    Dec 3 '18 at 8:56











    1












    $begingroup$

    It's an abelian group with nine elements, where each element has order $3$. There is only one such group (which I suppose is what you actually need to prove).






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      It's an abelian group with nine elements, where each element has order $3$. There is only one such group (which I suppose is what you actually need to prove).






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        It's an abelian group with nine elements, where each element has order $3$. There is only one such group (which I suppose is what you actually need to prove).






        share|cite|improve this answer









        $endgroup$



        It's an abelian group with nine elements, where each element has order $3$. There is only one such group (which I suppose is what you actually need to prove).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 3 '18 at 8:40









        ArthurArthur

        113k7110193




        113k7110193















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