Prove ring of dyadic rationals is a Euclidean domain

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This is a question from page 105, Vinberg - A course in Algebra:



Prove that the ring $A$ of rational numbers of the form $ 2^-nm,~m in mathbbZ,~n in mathbbZ_+ $, is a Euclidean domain.



Here $mathbbZ_+$ contains 0. I try to find a norm $N colon A setminus 0 to mathbbZ_+$ that satisfies Euclidean algorithm. I suppose that $N$ has to satisfy the following conditions:



  1. Since each rational number is an equivalent class, the norm $N$ must not depend on the representative of the equivalent class.


  2. $N$ somehow measures the "distance" between elements in A and 0 (as we did for the case Gaussian integers $mathbbZ[i]$) because we need to compare the norms of divisor and remainder.


However, I can't find any reasonable formula for $N$. Any ideas?




abstract-algebra euclidean-algorithm euclidean-domain




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    This is a question from page 105, Vinberg - A course in Algebra:



    Prove that the ring $A$ of rational numbers of the form $ 2^-nm,~m in mathbbZ,~n in mathbbZ_+ $, is a Euclidean domain.



    Here $mathbbZ_+$ contains 0. I try to find a norm $N colon A setminus 0 to mathbbZ_+$ that satisfies Euclidean algorithm. I suppose that $N$ has to satisfy the following conditions:



    1. Since each rational number is an equivalent class, the norm $N$ must not depend on the representative of the equivalent class.


    2. $N$ somehow measures the "distance" between elements in A and 0 (as we did for the case Gaussian integers $mathbbZ[i]$) because we need to compare the norms of divisor and remainder.


    However, I can't find any reasonable formula for $N$. Any ideas?




    abstract-algebra euclidean-algorithm euclidean-domain




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      This is a question from page 105, Vinberg - A course in Algebra:



      Prove that the ring $A$ of rational numbers of the form $ 2^-nm,~m in mathbbZ,~n in mathbbZ_+ $, is a Euclidean domain.



      Here $mathbbZ_+$ contains 0. I try to find a norm $N colon A setminus 0 to mathbbZ_+$ that satisfies Euclidean algorithm. I suppose that $N$ has to satisfy the following conditions:



      1. Since each rational number is an equivalent class, the norm $N$ must not depend on the representative of the equivalent class.


      2. $N$ somehow measures the "distance" between elements in A and 0 (as we did for the case Gaussian integers $mathbbZ[i]$) because we need to compare the norms of divisor and remainder.


      However, I can't find any reasonable formula for $N$. Any ideas?




      abstract-algebra euclidean-algorithm euclidean-domain




      share|cite|improve this question











      This is a question from page 105, Vinberg - A course in Algebra:



      Prove that the ring $A$ of rational numbers of the form $ 2^-nm,~m in mathbbZ,~n in mathbbZ_+ $, is a Euclidean domain.



      Here $mathbbZ_+$ contains 0. I try to find a norm $N colon A setminus 0 to mathbbZ_+$ that satisfies Euclidean algorithm. I suppose that $N$ has to satisfy the following conditions:



      1. Since each rational number is an equivalent class, the norm $N$ must not depend on the representative of the equivalent class.


      2. $N$ somehow measures the "distance" between elements in A and 0 (as we did for the case Gaussian integers $mathbbZ[i]$) because we need to compare the norms of divisor and remainder.


      However, I can't find any reasonable formula for $N$. Any ideas?





      abstract-algebra euclidean-algorithm euclidean-domain





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      asked Jul 22 at 18:07









      macnguyen

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          Each nonzero element in the ring is $2^ka$ where $kinBbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,






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            up vote
            1
            down vote



            accepted










            Each nonzero element in the ring is $2^ka$ where $kinBbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              Each nonzero element in the ring is $2^ka$ where $kinBbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,






              share|cite|improve this answer























                up vote
                1
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                accepted







                up vote
                1
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                accepted






                Each nonzero element in the ring is $2^ka$ where $kinBbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,






                share|cite|improve this answer













                Each nonzero element in the ring is $2^ka$ where $kinBbb Z$ and $a$ an odd integer. Define $N(2^ka)=|a|$,







                share|cite|improve this answer













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                answered Jul 22 at 18:13









                Lord Shark the Unknown

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