Upper bound for class number of field
$begingroup$
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).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
$endgroup$
add a comment |
$begingroup$
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).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
$endgroup$
add a comment |
$begingroup$
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).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
$endgroup$
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
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
asked Dec 4 '18 at 2:04
J. LinneJ. Linne
863315
863315
add a comment |
add a comment |
1 Answer
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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 ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
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1 Answer
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1 Answer
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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 ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
$endgroup$
add a comment |
$begingroup$
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 ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
$endgroup$
add a comment |
$begingroup$
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 ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
$endgroup$
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 ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
answered Dec 27 '18 at 18:46
franz lemmermeyerfranz lemmermeyer
7,13922047
7,13922047
add a comment |
add a comment |
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