Structure of unit group of algebraic integers $overline{mathbb{Z}}$
Dirichlet's unit theorem tells us that for any number field $K$, it is well known that the unit group $mathcal{O}_{K}^{times}$ of the ring of integers $mathcal{O}_{K}$ of $K$ is isomorphic to $mu(K)times mathbb{Z}^{r+s-1}$, where $mu(K)$ is a (finite) group of root of unities and free part corresponds to fundamental units.
My question is: is there any generalization of the theorem for infinite algebraic extensions? For example, let $overline{mathbb{Z}} = mathcal{O}_{overline{mathbb{Q}}}$ be a ring of algebraic integers. Can we describe $overline{mathbb{Z}}^{times}$? I think it should be $(mathbb{Q}/mathbb{Z})oplusleft(bigoplus mathbb{Q}right)$, where the direct sum has countably many copies of $mathbb{Q}$, because it may be a direct limit of $mathcal{O}_{K}^{times}$. Is this true? If it is true, can we write element of $overline{mathbb{Z}}^{times}$ that corresponds to each $mathbb{Q}$-summands? Thanks in advance.
algebraic-number-theory
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
add a comment |
Dirichlet's unit theorem tells us that for any number field $K$, it is well known that the unit group $mathcal{O}_{K}^{times}$ of the ring of integers $mathcal{O}_{K}$ of $K$ is isomorphic to $mu(K)times mathbb{Z}^{r+s-1}$, where $mu(K)$ is a (finite) group of root of unities and free part corresponds to fundamental units.
My question is: is there any generalization of the theorem for infinite algebraic extensions? For example, let $overline{mathbb{Z}} = mathcal{O}_{overline{mathbb{Q}}}$ be a ring of algebraic integers. Can we describe $overline{mathbb{Z}}^{times}$? I think it should be $(mathbb{Q}/mathbb{Z})oplusleft(bigoplus mathbb{Q}right)$, where the direct sum has countably many copies of $mathbb{Q}$, because it may be a direct limit of $mathcal{O}_{K}^{times}$. Is this true? If it is true, can we write element of $overline{mathbb{Z}}^{times}$ that corresponds to each $mathbb{Q}$-summands? Thanks in advance.
algebraic-number-theory
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
1
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22
add a comment |
Dirichlet's unit theorem tells us that for any number field $K$, it is well known that the unit group $mathcal{O}_{K}^{times}$ of the ring of integers $mathcal{O}_{K}$ of $K$ is isomorphic to $mu(K)times mathbb{Z}^{r+s-1}$, where $mu(K)$ is a (finite) group of root of unities and free part corresponds to fundamental units.
My question is: is there any generalization of the theorem for infinite algebraic extensions? For example, let $overline{mathbb{Z}} = mathcal{O}_{overline{mathbb{Q}}}$ be a ring of algebraic integers. Can we describe $overline{mathbb{Z}}^{times}$? I think it should be $(mathbb{Q}/mathbb{Z})oplusleft(bigoplus mathbb{Q}right)$, where the direct sum has countably many copies of $mathbb{Q}$, because it may be a direct limit of $mathcal{O}_{K}^{times}$. Is this true? If it is true, can we write element of $overline{mathbb{Z}}^{times}$ that corresponds to each $mathbb{Q}$-summands? Thanks in advance.
algebraic-number-theory
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
Dirichlet's unit theorem tells us that for any number field $K$, it is well known that the unit group $mathcal{O}_{K}^{times}$ of the ring of integers $mathcal{O}_{K}$ of $K$ is isomorphic to $mu(K)times mathbb{Z}^{r+s-1}$, where $mu(K)$ is a (finite) group of root of unities and free part corresponds to fundamental units.
My question is: is there any generalization of the theorem for infinite algebraic extensions? For example, let $overline{mathbb{Z}} = mathcal{O}_{overline{mathbb{Q}}}$ be a ring of algebraic integers. Can we describe $overline{mathbb{Z}}^{times}$? I think it should be $(mathbb{Q}/mathbb{Z})oplusleft(bigoplus mathbb{Q}right)$, where the direct sum has countably many copies of $mathbb{Q}$, because it may be a direct limit of $mathcal{O}_{K}^{times}$. Is this true? If it is true, can we write element of $overline{mathbb{Z}}^{times}$ that corresponds to each $mathbb{Q}$-summands? Thanks in advance.
algebraic-number-theory
algebraic-number-theory
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
asked Nov 22 '18 at 22:25
Seewoo LeeSeewoo Lee
6,232826
6,232826
The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
1
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22
add a comment |
The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
1
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22
The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
1
1
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22
add a comment |
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The structure sounds right to me, $overline{mathbf Z}^times$ should be a divisible group with torsion part the set of all roots of unity, so the structure follows from this also. The second part of your question seems far harder to do in a natural way, if we write down such an isomorphism with $bigoplus mathbf Q$ we probably had to make a lot of choices along the way, in the order we considered the individual number fields at least.
– Alex J Best
Nov 22 '18 at 23:03
@AlexJBest Thanks. I also think that it is impossible to find canonical elements correspond to each $mathbb{Q}$. By the way, is it possible to find $N$ elements in $overline{mathbb{Z}}^{times}$ such that they are $mathbb{Q}$-linearly independent (with respect to multiplication) for any $N$? For example, I strongly believe that $1+sqrt{2}$ and $2+sqrt{3}$ are $mathbb{Q}$-linearly independent, i.e. $(1+sqrt{2})^{a}(2+sqrt{3})^{b}neq 1$ for any $(0, 0)neq (a, b)in mathbb{Q}^{2}$.
– Seewoo Lee
Nov 22 '18 at 23:19
@reuns Do you have any suggestions for that unit group? I also thought about that too.
– Seewoo Lee
Nov 22 '18 at 23:20
1
There is probably more to say about $mathcal{O}_{mathbb{Q}(zeta_infty)}^times$ (the $log$ of the Minkowski embedding of $mathcal{O}_K^times$ is some lattice in some real subspace of $mathbb{C}^{n}$ and we can try understanding how those vary with field extensions)
– reuns
Nov 22 '18 at 23:22