Find the set of interstion of $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ shwoing its elements...

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  • Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]

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This question is from $text{p-adic numbers}.$



My questions are-



$(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? If non-empty, then what are the elements or the intersection set?



$(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty? If non-empty what are the elements or the intersection set?



I can not conclude the answer.



Please someone help me with details answer or at least hints.






p-adic-number-theory







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marked as duplicate by Torsten Schoeneberg, Ali Caglayan, Rebellos, José Carlos Santos, supinf Nov 29 at 14:41


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.



















    up vote
    1
    down vote

    favorite
    2













    This question already has an answer here:




    • Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]

      1 answer




    This question is from $text{p-adic numbers}.$



    My questions are-



    $(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? If non-empty, then what are the elements or the intersection set?



    $(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty? If non-empty what are the elements or the intersection set?



    I can not conclude the answer.



    Please someone help me with details answer or at least hints.






    p-adic-number-theory







    share|cite|improve this question













    marked as duplicate by Torsten Schoeneberg, Ali Caglayan, Rebellos, José Carlos Santos, supinf Nov 29 at 14:41


    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.

















      up vote
      1
      down vote

      favorite
      2









      up vote
      1
      down vote

      favorite
      2






      2






      This question already has an answer here:




      • Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]

        1 answer




      This question is from $text{p-adic numbers}.$



      My questions are-



      $(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? If non-empty, then what are the elements or the intersection set?



      $(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty? If non-empty what are the elements or the intersection set?



      I can not conclude the answer.



      Please someone help me with details answer or at least hints.






      p-adic-number-theory







      share|cite|improve this question














      This question already has an answer here:




      • Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]

        1 answer




      This question is from $text{p-adic numbers}.$



      My questions are-



      $(1)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? If non-empty, then what are the elements or the intersection set?



      $(2)$ Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}_p $ non-empty? If non-empty what are the elements or the intersection set?



      I can not conclude the answer.



      Please someone help me with details answer or at least hints.





      This question already has an answer here:




      • Is the set $ (mathbb{Z}_p setminus mathbb{Z}) cap mathbb{Q}$ non-empty? [closed]

        1 answer







      p-adic-number-theory




      p-adic-number-theory






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      share|cite|improve this question











      share|cite|improve this question




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      asked Nov 29 at 4:08









      M. A. SARKAR

      2,1271619




      2,1271619




      marked as duplicate by Torsten Schoeneberg, Ali Caglayan, Rebellos, José Carlos Santos, supinf Nov 29 at 14:41


      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.






      marked as duplicate by Torsten Schoeneberg, Ali Caglayan, Rebellos, José Carlos Santos, supinf Nov 29 at 14:41


      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 Answer
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          Hint: $frac{1}{2} in mathbb{Z}_3 setminus mathbb{Z}$, since $$frac{1}{2}= frac{-1}{1-3} = -1-3-9-27- dots.$$ That (1) is nonempty implies that (2) is.



          Similarly, you can prove that any $frac{a}{b} in mathbb{Z}_3 setminus mathbb{Z}$ so long $3 nmid b.$






          share|cite|improve this answer























          • yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
            – M. A. SARKAR
            Nov 29 at 4:17












          • Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
            – Dzoooks
            Nov 29 at 4:19










          • Does $ frac{a}{b} in mathbb{Q}_p$?
            – M. A. SARKAR
            Nov 29 at 4:19






          • 1




            Yes. $mathbb{Q}_p supset mathbb{Q}$.
            – Dzoooks
            Nov 29 at 4:20






          • 1




            Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
            – KCd
            Nov 29 at 6:28


















          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          1
          down vote













          Hint: $frac{1}{2} in mathbb{Z}_3 setminus mathbb{Z}$, since $$frac{1}{2}= frac{-1}{1-3} = -1-3-9-27- dots.$$ That (1) is nonempty implies that (2) is.



          Similarly, you can prove that any $frac{a}{b} in mathbb{Z}_3 setminus mathbb{Z}$ so long $3 nmid b.$






          share|cite|improve this answer























          • yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
            – M. A. SARKAR
            Nov 29 at 4:17












          • Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
            – Dzoooks
            Nov 29 at 4:19










          • Does $ frac{a}{b} in mathbb{Q}_p$?
            – M. A. SARKAR
            Nov 29 at 4:19






          • 1




            Yes. $mathbb{Q}_p supset mathbb{Q}$.
            – Dzoooks
            Nov 29 at 4:20






          • 1




            Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
            – KCd
            Nov 29 at 6:28















          up vote
          1
          down vote













          Hint: $frac{1}{2} in mathbb{Z}_3 setminus mathbb{Z}$, since $$frac{1}{2}= frac{-1}{1-3} = -1-3-9-27- dots.$$ That (1) is nonempty implies that (2) is.



          Similarly, you can prove that any $frac{a}{b} in mathbb{Z}_3 setminus mathbb{Z}$ so long $3 nmid b.$






          share|cite|improve this answer























          • yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
            – M. A. SARKAR
            Nov 29 at 4:17












          • Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
            – Dzoooks
            Nov 29 at 4:19










          • Does $ frac{a}{b} in mathbb{Q}_p$?
            – M. A. SARKAR
            Nov 29 at 4:19






          • 1




            Yes. $mathbb{Q}_p supset mathbb{Q}$.
            – Dzoooks
            Nov 29 at 4:20






          • 1




            Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
            – KCd
            Nov 29 at 6:28













          up vote
          1
          down vote










          up vote
          1
          down vote









          Hint: $frac{1}{2} in mathbb{Z}_3 setminus mathbb{Z}$, since $$frac{1}{2}= frac{-1}{1-3} = -1-3-9-27- dots.$$ That (1) is nonempty implies that (2) is.



          Similarly, you can prove that any $frac{a}{b} in mathbb{Z}_3 setminus mathbb{Z}$ so long $3 nmid b.$






          share|cite|improve this answer














          Hint: $frac{1}{2} in mathbb{Z}_3 setminus mathbb{Z}$, since $$frac{1}{2}= frac{-1}{1-3} = -1-3-9-27- dots.$$ That (1) is nonempty implies that (2) is.



          Similarly, you can prove that any $frac{a}{b} in mathbb{Z}_3 setminus mathbb{Z}$ so long $3 nmid b.$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 29 at 4:17

























          answered Nov 29 at 4:15









          Dzoooks

          846316




          846316












          • yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
            – M. A. SARKAR
            Nov 29 at 4:17












          • Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
            – Dzoooks
            Nov 29 at 4:19










          • Does $ frac{a}{b} in mathbb{Q}_p$?
            – M. A. SARKAR
            Nov 29 at 4:19






          • 1




            Yes. $mathbb{Q}_p supset mathbb{Q}$.
            – Dzoooks
            Nov 29 at 4:20






          • 1




            Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
            – KCd
            Nov 29 at 6:28


















          • yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
            – M. A. SARKAR
            Nov 29 at 4:17












          • Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
            – Dzoooks
            Nov 29 at 4:19










          • Does $ frac{a}{b} in mathbb{Q}_p$?
            – M. A. SARKAR
            Nov 29 at 4:19






          • 1




            Yes. $mathbb{Q}_p supset mathbb{Q}$.
            – Dzoooks
            Nov 29 at 4:20






          • 1




            Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
            – KCd
            Nov 29 at 6:28
















          yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
          – M. A. SARKAR
          Nov 29 at 4:17






          yes that is one element and so non-empty.. But I need to find the whole and general expression of the intersection set .. Can you do that that for general $p$?
          – M. A. SARKAR
          Nov 29 at 4:17














          Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
          – Dzoooks
          Nov 29 at 4:19




          Simply inductive construct the inverse power series of $b$ for $frac{1}{b}.$
          – Dzoooks
          Nov 29 at 4:19












          Does $ frac{a}{b} in mathbb{Q}_p$?
          – M. A. SARKAR
          Nov 29 at 4:19




          Does $ frac{a}{b} in mathbb{Q}_p$?
          – M. A. SARKAR
          Nov 29 at 4:19




          1




          1




          Yes. $mathbb{Q}_p supset mathbb{Q}$.
          – Dzoooks
          Nov 29 at 4:20




          Yes. $mathbb{Q}_p supset mathbb{Q}$.
          – Dzoooks
          Nov 29 at 4:20




          1




          1




          Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
          – KCd
          Nov 29 at 6:28




          Asking someone to put an answer in that special form sounds like you are asking someone to provide the answer to a homework question. That is not a reasonable request.
          – KCd
          Nov 29 at 6:28



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          -
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