Upper bound for class number of field












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$begingroup$


Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.



Example of conjecture above:



The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).










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    2












    $begingroup$


    Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.



    Example of conjecture above:



    The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).










    share|cite|improve this question









    $endgroup$















      2












      2








      2


      0



      $begingroup$


      Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.



      Example of conjecture above:



      The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).










      share|cite|improve this question









      $endgroup$




      Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.



      Example of conjecture above:



      The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).







      elementary-number-theory class-field-theory






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      asked Dec 4 '18 at 2:04









      J. LinneJ. Linne

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          $begingroup$

          Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.






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            $begingroup$

            Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.






                share|cite|improve this answer









                $endgroup$



                Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.







                share|cite|improve this answer












                share|cite|improve this answer



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                answered Dec 27 '18 at 18:46









                franz lemmermeyerfranz lemmermeyer

                7,07522047




                7,07522047






























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