Mixing
Property of quotient ideals in a number Field
Clash Royale CLAN TAG#URR8PPP
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Let $mathbbK$ be an algebraic number field.
Let $I$ and $J$ be two fractional ideals of $mathbbK$ and $q in mathbbN$ a positive integer.
Is it true that $I/qI $ isomorphic to $J/qJ$? If yes, what is the isomorphism?
If no, is there a condition on $q$ that make it True?
linear-algebra ring-theory field-theory algebraic-number-theory ideals
asked Jul 20 at 13:47
C.S.
134
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show 1 more comment
up vote
0
down vote
favorite
Let $mathbbK$ be an algebraic number field.
Let $I$ and $J$ be two fractional ideals of $mathbbK$ and $q in mathbbN$ a positive integer.
Is it true that $I/qI $ isomorphic to $J/qJ$? If yes, what is the isomorphism?
If no, is there a condition on $q$ that make it True?
linear-algebra ring-theory field-theory algebraic-number-theory ideals
asked Jul 20 at 13:47
C.S.
134
Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23
 |Â
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $mathbbK$ be an algebraic number field.
Let $I$ and $J$ be two fractional ideals of $mathbbK$ and $q in mathbbN$ a positive integer.
Is it true that $I/qI $ isomorphic to $J/qJ$? If yes, what is the isomorphism?
If no, is there a condition on $q$ that make it True?
linear-algebra ring-theory field-theory algebraic-number-theory ideals
asked Jul 20 at 13:47
C.S.
134
Let $mathbbK$ be an algebraic number field.
Let $I$ and $J$ be two fractional ideals of $mathbbK$ and $q in mathbbN$ a positive integer.
Is it true that $I/qI $ isomorphic to $J/qJ$? If yes, what is the isomorphism?
If no, is there a condition on $q$ that make it True?
linear-algebra ring-theory field-theory algebraic-number-theory ideals
asked Jul 20 at 13:47
C.S.
134
asked Jul 20 at 13:47
C.S.
134
asked Jul 20 at 13:47
C.S.
134
asked Jul 20 at 13:47
C.S.
134
134
Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23
 |Â
show 1 more comment
Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23
Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23
 |Â
show 1 more comment
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Isomorphic in what sense? As abelian groups? As $mathcalO_mathbbK$-modules?
– Gal Porat
Jul 20 at 15:21
Besides, it will certainly not be true in any reasonable sense if you take $mathbbK = mathbbQ$, $I=(0), J=(1)$
– Gal Porat
Jul 20 at 15:22
What happen If $mathbbK$ is a cyclotomic number field?, and the ideals are not trivials.
– C.S.
Jul 20 at 16:37
But you have still not specified what you mean by your question...
– Gal Porat
Jul 20 at 17:48
No condition depending only on $q$ seems to exist as both $I/qI$ and $J/qJ$ are annihilated by $q$. There must be conditions on $I$ and $J$ to have isomorphism in certain circumstances.
– joy
Jul 23 at 8:23