Consider the order $mathbb{Z}[sqrt[4]{24}]$. Find all ideals of norm 100.












2












$begingroup$


I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?



I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.










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$endgroup$












  • $begingroup$
    Does that help? What do you mean?
    $endgroup$
    – Rubisk
    Jan 16 at 19:01










  • $begingroup$
    Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
    $endgroup$
    – Rubisk
    Jan 19 at 21:38












  • $begingroup$
    Assuming the class group is trivial?
    $endgroup$
    – Rubisk
    Jan 19 at 21:57
















2












$begingroup$


I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?



I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Does that help? What do you mean?
    $endgroup$
    – Rubisk
    Jan 16 at 19:01










  • $begingroup$
    Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
    $endgroup$
    – Rubisk
    Jan 19 at 21:38












  • $begingroup$
    Assuming the class group is trivial?
    $endgroup$
    – Rubisk
    Jan 19 at 21:57














2












2








2





$begingroup$


I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?



I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.










share|cite|improve this question











$endgroup$




I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?



I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.







algebraic-number-theory norm ideals






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 19 at 21:39







Rubisk

















asked Jan 16 at 17:02









RubiskRubisk

112




112












  • $begingroup$
    Does that help? What do you mean?
    $endgroup$
    – Rubisk
    Jan 16 at 19:01










  • $begingroup$
    Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
    $endgroup$
    – Rubisk
    Jan 19 at 21:38












  • $begingroup$
    Assuming the class group is trivial?
    $endgroup$
    – Rubisk
    Jan 19 at 21:57


















  • $begingroup$
    Does that help? What do you mean?
    $endgroup$
    – Rubisk
    Jan 16 at 19:01










  • $begingroup$
    Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
    $endgroup$
    – Rubisk
    Jan 19 at 21:38












  • $begingroup$
    Assuming the class group is trivial?
    $endgroup$
    – Rubisk
    Jan 19 at 21:57
















$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01




$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01












$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38






$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38














$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57




$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57










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