Why $R/(a+bi)$ is a finite field with characteristic $p$ where $a^2+b^2=p$ [duplicate]












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  • Quotient ring of Gaussian integers

    6 answers




Suppose $R=mathbb{Z}[i]$ is Gaussian domain , and $a^2+b^2=p$ is prime. Denote $alpha=a+bi$ and $I=(alpha)$. Prove $R/I$ is a finite field with characteristic p.



My attempts: First prove $alpha $ is irreducible, hence $I$ is a maximal ideal and it follows $R/I$ is a field. And $R=mathbb{Z}[i]$ is an Euclidean domain. Hence for $xin mathbb{Z}[i],x=qalpha+r$ where $N(r)lt N(alpha)$ , so there are finite elements in $R/I$. Thus $R/I$ is a finite filed. But I can't figure out why the characteristic is $p$.



Thanks for your hints.






abstract-algebra finite-fields







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 2 days ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.











  • 1




    $R/I$ has $p$ elements.
    – Lord Shark the Unknown
    2 days ago










  • always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
    – Mustafa
    2 days ago










  • How to determine the number of elements in $R/I$?
    – Jaqen Chou
    2 days ago
















0















This question already has an answer here:




  • Quotient ring of Gaussian integers

    6 answers




Suppose $R=mathbb{Z}[i]$ is Gaussian domain , and $a^2+b^2=p$ is prime. Denote $alpha=a+bi$ and $I=(alpha)$. Prove $R/I$ is a finite field with characteristic p.



My attempts: First prove $alpha $ is irreducible, hence $I$ is a maximal ideal and it follows $R/I$ is a field. And $R=mathbb{Z}[i]$ is an Euclidean domain. Hence for $xin mathbb{Z}[i],x=qalpha+r$ where $N(r)lt N(alpha)$ , so there are finite elements in $R/I$. Thus $R/I$ is a finite filed. But I can't figure out why the characteristic is $p$.



Thanks for your hints.






abstract-algebra finite-fields







share|cite|improve this question













marked as duplicate by André 3000, amWhy abstract-algebra
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 2 days ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.











  • 1




    $R/I$ has $p$ elements.
    – Lord Shark the Unknown
    2 days ago










  • always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
    – Mustafa
    2 days ago










  • How to determine the number of elements in $R/I$?
    – Jaqen Chou
    2 days ago














0












0








0








This question already has an answer here:




  • Quotient ring of Gaussian integers

    6 answers




Suppose $R=mathbb{Z}[i]$ is Gaussian domain , and $a^2+b^2=p$ is prime. Denote $alpha=a+bi$ and $I=(alpha)$. Prove $R/I$ is a finite field with characteristic p.



My attempts: First prove $alpha $ is irreducible, hence $I$ is a maximal ideal and it follows $R/I$ is a field. And $R=mathbb{Z}[i]$ is an Euclidean domain. Hence for $xin mathbb{Z}[i],x=qalpha+r$ where $N(r)lt N(alpha)$ , so there are finite elements in $R/I$. Thus $R/I$ is a finite filed. But I can't figure out why the characteristic is $p$.



Thanks for your hints.






abstract-algebra finite-fields







share|cite|improve this question














This question already has an answer here:




  • Quotient ring of Gaussian integers

    6 answers




Suppose $R=mathbb{Z}[i]$ is Gaussian domain , and $a^2+b^2=p$ is prime. Denote $alpha=a+bi$ and $I=(alpha)$. Prove $R/I$ is a finite field with characteristic p.



My attempts: First prove $alpha $ is irreducible, hence $I$ is a maximal ideal and it follows $R/I$ is a field. And $R=mathbb{Z}[i]$ is an Euclidean domain. Hence for $xin mathbb{Z}[i],x=qalpha+r$ where $N(r)lt N(alpha)$ , so there are finite elements in $R/I$. Thus $R/I$ is a finite filed. But I can't figure out why the characteristic is $p$.



Thanks for your hints.





This question already has an answer here:




  • Quotient ring of Gaussian integers

    6 answers







abstract-algebra finite-fields




abstract-algebra finite-fields






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









Jaqen Chou

409110




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marked as duplicate by André 3000, amWhy abstract-algebra
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This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.










  • 1




    $R/I$ has $p$ elements.
    – Lord Shark the Unknown
    2 days ago










  • always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
    – Mustafa
    2 days ago










  • How to determine the number of elements in $R/I$?
    – Jaqen Chou
    2 days ago














  • 1




    $R/I$ has $p$ elements.
    – Lord Shark the Unknown
    2 days ago










  • always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
    – Mustafa
    2 days ago










  • How to determine the number of elements in $R/I$?
    – Jaqen Chou
    2 days ago








1




1




$R/I$ has $p$ elements.
– Lord Shark the Unknown
2 days ago




$R/I$ has $p$ elements.
– Lord Shark the Unknown
2 days ago












always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
– Mustafa
2 days ago




always $mathbb Z[i]/(a+bi)cong mathbb Z/N(a+bi)mathbb Z$ where $(a,b)=1$
– Mustafa
2 days ago












How to determine the number of elements in $R/I$?
– Jaqen Chou
2 days ago




How to determine the number of elements in $R/I$?
– Jaqen Chou
2 days ago










1 Answer
1






active

oldest

votes


















2














$p = a^2+b^2 = bar alpha alpha in I$ implies $p=0$ in $R/I$.



If you know that $R/I$ is a field, then it must have characteristic $p$.






share|cite|improve this answer




























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2














    $p = a^2+b^2 = bar alpha alpha in I$ implies $p=0$ in $R/I$.



    If you know that $R/I$ is a field, then it must have characteristic $p$.






    share|cite|improve this answer


























      2














      $p = a^2+b^2 = bar alpha alpha in I$ implies $p=0$ in $R/I$.



      If you know that $R/I$ is a field, then it must have characteristic $p$.






      share|cite|improve this answer
























        2












        2








        2






        $p = a^2+b^2 = bar alpha alpha in I$ implies $p=0$ in $R/I$.



        If you know that $R/I$ is a field, then it must have characteristic $p$.






        share|cite|improve this answer












        $p = a^2+b^2 = bar alpha alpha in I$ implies $p=0$ in $R/I$.



        If you know that $R/I$ is a field, then it must have characteristic $p$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 days ago









        lhf

        163k10167387




        163k10167387















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