Cyclotomic fields and splitting of central simple algebras
$begingroup$
Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $mathbb{Q}$-algebra embedding $K hookrightarrow A$?
This is equivalent to asking whether $A otimes_{mathbb{Q}} K simeq M_n(K)$. Is there a local-to-global principle that can be used?
If, for example, $A simeq M_n(D)$ where $D$ is a quaternion algebra over $mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?
Any reference to a similar example where this is worked out would be greatly appreciated!
nt.number-theory ra.rings-and-algebras central-simple-algebras
asked Jan 28 at 14:23
Sun RaSun Ra
532
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add a comment |
$begingroup$
Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $mathbb{Q}$-algebra embedding $K hookrightarrow A$?
This is equivalent to asking whether $A otimes_{mathbb{Q}} K simeq M_n(K)$. Is there a local-to-global principle that can be used?
If, for example, $A simeq M_n(D)$ where $D$ is a quaternion algebra over $mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?
Any reference to a similar example where this is worked out would be greatly appreciated!
nt.number-theory ra.rings-and-algebras central-simple-algebras
asked Jan 28 at 14:23
Sun RaSun Ra
532
$endgroup$
add a comment |
$begingroup$
Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $mathbb{Q}$-algebra embedding $K hookrightarrow A$?
This is equivalent to asking whether $A otimes_{mathbb{Q}} K simeq M_n(K)$. Is there a local-to-global principle that can be used?
If, for example, $A simeq M_n(D)$ where $D$ is a quaternion algebra over $mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?
Any reference to a similar example where this is worked out would be greatly appreciated!
nt.number-theory ra.rings-and-algebras central-simple-algebras
asked Jan 28 at 14:23
Sun RaSun Ra
532
$endgroup$
Let $K$ be a cyclotomic field of degree $n$ and $A$ a central simple algebra over $mathbb{Q}$ of dimension $n^2$. How can one determine whether there is a $mathbb{Q}$-algebra embedding $K hookrightarrow A$?
This is equivalent to asking whether $A otimes_{mathbb{Q}} K simeq M_n(K)$. Is there a local-to-global principle that can be used?
If, for example, $A simeq M_n(D)$ where $D$ is a quaternion algebra over $mathbb{Q}$, is there a simple criterion relating subfields of $K$ and splitting fields of $D$?
Any reference to a similar example where this is worked out would be greatly appreciated!
nt.number-theory ra.rings-and-algebras central-simple-algebras
nt.number-theory ra.rings-and-algebras central-simple-algebras
asked Jan 28 at 14:23
Sun RaSun Ra
532
asked Jan 28 at 14:23
Sun RaSun Ra
532
asked Jan 28 at 14:23
Sun RaSun Ra
532
asked Jan 28 at 14:23
Sun RaSun Ra
532
asked Jan 28 at 14:23
Sun RaSun Ra
532
532
add a comment |
add a comment |
1 Answer
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I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $mathfrak p$ of $F$ (possibly infinite) denote by $m_frak{p}$ the local index of $A$ at $mathfrak{p}$ (i.e., the degree of the division algebra part of $Aotimes_F F_mathfrak{p}$).
The following is Theorem 32.15 of Reiner.
Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $mathfrak p$ of $F$ and prime $mathfrak P$ of $L$ lying above $mathfrak p$, we have:
begin{equation}
m_{mathfrak p}mid [L_mathfrak P:F_mathfrak p].
end{equation}
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
$endgroup$
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $mathfrak p$ of $F$ (possibly infinite) denote by $m_frak{p}$ the local index of $A$ at $mathfrak{p}$ (i.e., the degree of the division algebra part of $Aotimes_F F_mathfrak{p}$).
The following is Theorem 32.15 of Reiner.
Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $mathfrak p$ of $F$ and prime $mathfrak P$ of $L$ lying above $mathfrak p$, we have:
begin{equation}
m_{mathfrak p}mid [L_mathfrak P:F_mathfrak p].
end{equation}
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
$endgroup$
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
add a comment |
$begingroup$
I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $mathfrak p$ of $F$ (possibly infinite) denote by $m_frak{p}$ the local index of $A$ at $mathfrak{p}$ (i.e., the degree of the division algebra part of $Aotimes_F F_mathfrak{p}$).
The following is Theorem 32.15 of Reiner.
Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $mathfrak p$ of $F$ and prime $mathfrak P$ of $L$ lying above $mathfrak p$, we have:
begin{equation}
m_{mathfrak p}mid [L_mathfrak P:F_mathfrak p].
end{equation}
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
$endgroup$
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
add a comment |
$begingroup$
I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $mathfrak p$ of $F$ (possibly infinite) denote by $m_frak{p}$ the local index of $A$ at $mathfrak{p}$ (i.e., the degree of the division algebra part of $Aotimes_F F_mathfrak{p}$).
The following is Theorem 32.15 of Reiner.
Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $mathfrak p$ of $F$ and prime $mathfrak P$ of $L$ lying above $mathfrak p$, we have:
begin{equation}
m_{mathfrak p}mid [L_mathfrak P:F_mathfrak p].
end{equation}
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
$endgroup$
I believe that the characterization you are after is a more or less straightforward consequence of the Albert-Brauer-Hasse-Noether theorem and recommend that you take a look at Chapter 32 of Reiner's book Maximal Orders. The chapter is entitled "Splitting of Simple Algebras" and covers the subject in great detail.
Let $F$ be a number field and $A$ be a finite dimensional central simple algebra over $F$. Given a prime $mathfrak p$ of $F$ (possibly infinite) denote by $m_frak{p}$ the local index of $A$ at $mathfrak{p}$ (i.e., the degree of the division algebra part of $Aotimes_F F_mathfrak{p}$).
The following is Theorem 32.15 of Reiner.
Theorem. Let $L$ be a finite extension of $F$. Then $L$ is a splitting field for $A$ if and only if for every prime $mathfrak p$ of $F$ and prime $mathfrak P$ of $L$ lying above $mathfrak p$, we have:
begin{equation}
m_{mathfrak p}mid [L_mathfrak P:F_mathfrak p].
end{equation}
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
edited Jan 29 at 13:20
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
answered Jan 28 at 14:48
Ben LinowitzBen Linowitz
5,96913255
5,96913255
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
add a comment |
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
Perhaps I am misunderstanding the definition of local index, but if I take $A/mathbb{Q}$ a quaternion algebra ramified at $p$, then $m_p=4$. So for a quadratic extension $L$ to split $A$, we would need $4|[L_mathfrak{p}:mathbb{Q}]$ which is impossible. Sorry for the confusion, I must be missing something.
$endgroup$
– Sun Ra
Feb 5 at 14:52
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
@SunRa - If $A$ is a rational quaternion algebra ramified at $p$ then the local index of $A$ at $p$ is $2=sqrt{4}=sqrt{dim(Aotimes_{mathbb Q} mathbb Q_p)}$. In general the local index is the degree (=square root of dimension) of the division algebra part of the central simple algebra, i.e., the division algebra $D$ for which your algebra is of the form $M_n(D)$.
$endgroup$
– Ben Linowitz
Feb 5 at 15:30
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
$begingroup$
That makes a lot more sense. Thanks, this is just the sort of criterion I was looking for.
$endgroup$
– Sun Ra
Feb 5 at 17:28
add a comment |
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