Orbits of the action of $mathrmSL_n (mathcalO_K)$ on $mathbbP^n-1(K)$ The 2019 Stack Overflow Developer Survey Results Are InEvery ideal of an algebraic number field can be principal in a suitable finite extension fieldQuadratic fields with cyclic class groupOrbits of the $textSL(n,mathcalO_K)$-action on $mathbbP^n-1(K)$ for a number field $K$.Class number of $mathbbQ(sqrt6)$Is the ratio of the norms of generators in an ideal well defined?$p$ splits completely in $mathbbQ(zeta_3,sqrt[3]2) Leftrightarrow p=x^2+27y^2$Index of Principal Ideal in a Subring of $mathbbQ(sqrt-3)$Calculating class numbersStandard argument for making the class group of a number field trivialGeneralized Principal ideal theorem for ray class groups
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Orbits of the action of $mathrmSL_n (mathcalO_K)$ on $mathbbP^n-1(K)$
The 2019 Stack Overflow Developer Survey Results Are InEvery ideal of an algebraic number field can be principal in a suitable finite extension fieldQuadratic fields with cyclic class groupOrbits of the $textSL(n,mathcalO_K)$-action on $mathbbP^n-1(K)$ for a number field $K$.Class number of $mathbbQ(sqrt6)$Is the ratio of the norms of generators in an ideal well defined?$p$ splits completely in $mathbbQ(zeta_3,sqrt[3]2) Leftrightarrow p=x^2+27y^2$Index of Principal Ideal in a Subring of $mathbbQ(sqrt-3)$Calculating class numbersStandard argument for making the class group of a number field trivialGeneralized Principal ideal theorem for ray class groups
$begingroup$
Let $K$ be a number field, and let $mathcalO_K$ be the ring of integers of $K$. Consider the natural action of $mathrmSL_n ( mathcalO_K)$ on $mathbbP^n-1(K)$. It is not difficult to show that if $mathcalO_K$ is a principal ideal domain, this action is transitive.
Is there a generalization of this fact to the case when the class group is not trivial? To be precise, if I have a point $P=(a_0:a_1:...a_n-1)$, is there a way to explicitly compute a matrix $g in mathrmSL_n ( mathcalO_K)$ so that the point $g cdot (a_0:a_1:...a_n-1)$ has "small" coordinates?
In the case $K=mathbbQ$, I can use Euclid's algorithm to compute a $g$ so that $g cdot (a_0:a_1:...a_n-1) = (1:0:0:...:0)$.
number-theory algebraic-number-theory computational-number-theory
asked Mar 23 at 16:43
fedlemmingfedlemming
173
$endgroup$
add a comment |
$begingroup$
Let $K$ be a number field, and let $mathcalO_K$ be the ring of integers of $K$. Consider the natural action of $mathrmSL_n ( mathcalO_K)$ on $mathbbP^n-1(K)$. It is not difficult to show that if $mathcalO_K$ is a principal ideal domain, this action is transitive.
Is there a generalization of this fact to the case when the class group is not trivial? To be precise, if I have a point $P=(a_0:a_1:...a_n-1)$, is there a way to explicitly compute a matrix $g in mathrmSL_n ( mathcalO_K)$ so that the point $g cdot (a_0:a_1:...a_n-1)$ has "small" coordinates?
In the case $K=mathbbQ$, I can use Euclid's algorithm to compute a $g$ so that $g cdot (a_0:a_1:...a_n-1) = (1:0:0:...:0)$.
number-theory algebraic-number-theory computational-number-theory
asked Mar 23 at 16:43
fedlemmingfedlemming
173
$endgroup$
$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19
add a comment |
$begingroup$
Let $K$ be a number field, and let $mathcalO_K$ be the ring of integers of $K$. Consider the natural action of $mathrmSL_n ( mathcalO_K)$ on $mathbbP^n-1(K)$. It is not difficult to show that if $mathcalO_K$ is a principal ideal domain, this action is transitive.
Is there a generalization of this fact to the case when the class group is not trivial? To be precise, if I have a point $P=(a_0:a_1:...a_n-1)$, is there a way to explicitly compute a matrix $g in mathrmSL_n ( mathcalO_K)$ so that the point $g cdot (a_0:a_1:...a_n-1)$ has "small" coordinates?
In the case $K=mathbbQ$, I can use Euclid's algorithm to compute a $g$ so that $g cdot (a_0:a_1:...a_n-1) = (1:0:0:...:0)$.
number-theory algebraic-number-theory computational-number-theory
asked Mar 23 at 16:43
fedlemmingfedlemming
173
$endgroup$
Let $K$ be a number field, and let $mathcalO_K$ be the ring of integers of $K$. Consider the natural action of $mathrmSL_n ( mathcalO_K)$ on $mathbbP^n-1(K)$. It is not difficult to show that if $mathcalO_K$ is a principal ideal domain, this action is transitive.
Is there a generalization of this fact to the case when the class group is not trivial? To be precise, if I have a point $P=(a_0:a_1:...a_n-1)$, is there a way to explicitly compute a matrix $g in mathrmSL_n ( mathcalO_K)$ so that the point $g cdot (a_0:a_1:...a_n-1)$ has "small" coordinates?
In the case $K=mathbbQ$, I can use Euclid's algorithm to compute a $g$ so that $g cdot (a_0:a_1:...a_n-1) = (1:0:0:...:0)$.
number-theory algebraic-number-theory computational-number-theory
number-theory algebraic-number-theory computational-number-theory
asked Mar 23 at 16:43
fedlemmingfedlemming
173
asked Mar 23 at 16:43
fedlemmingfedlemming
173
asked Mar 23 at 16:43
fedlemmingfedlemming
173
asked Mar 23 at 16:43
fedlemmingfedlemming
173
asked Mar 23 at 16:43
fedlemmingfedlemming
173
173
$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19
add a comment |
$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19
$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19
$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19
add a comment |
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$begingroup$
The orbits are in bijection with the class group by sending the point $P$ to the fractional ideal (well-defined up to multiplication by a principal ideal) $I = (a_0,a_1,ldots,a_n-1)$. (This implicitly uses that all ideals in the class group of number field can be generated by two elements.) So your question is very close to asking for algorithms to compute class groups as well as bounds for representatives for ideal classes. For the latter, you can use Minkowksi's bounds.
$endgroup$
– user655377
Mar 23 at 19:19