Mixing
Norms on cyclic division algebras over cyclic extensions
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Let $E=F(u^{1/p})/F$ be a cyclic field extension of odd prime degree $p$, $D$ a cyclic central cyclic division algebra over $F$, and assume that $D_E$ is still a division algebra. We have a norm homomorphism $N:Eto F$, which induces another norm map $N:D_Eto D$.
I am trying to understand the quotient group $D^*/N(D_E^*)$, and more specifically how it behaves wrt base change.
Question 1: Let $Lsubseteq D$ be a maximal subfield of $D$. Is it true that $L^*/N(L_E^*)hookrightarrow D^*/N(D_E^*)$? That is, do we have $N(D_E)cap L=N(L_E)$? (Probably not...).
A positive answer to question 1 would give a some grip on the following question.
Question 2: consider an integer $n$ with the following property. Let $xin D$, and assume that for some big field $Ksupseteq F$, there exists $yin D_{Eotimes K}$ such that $N(y)=x$ in $D_K$. Then there exists a subfield $K_0subseteq K$ such that $x$ is a norm already over $K_0$ (i.e. there is $zin D_{Etimes K_0}$ such that $x=N(z)$ in $D_{K_0}$). What is the smallest $n$ with this property?
If this makes computations easier (though I don't think it will), it is OK to assume $p=3$.
abstract-algebra galois-theory division-algebras
asked 14 hours ago
Cehiju
364
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show 3 more comments
up vote
2
down vote
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Let $E=F(u^{1/p})/F$ be a cyclic field extension of odd prime degree $p$, $D$ a cyclic central cyclic division algebra over $F$, and assume that $D_E$ is still a division algebra. We have a norm homomorphism $N:Eto F$, which induces another norm map $N:D_Eto D$.
I am trying to understand the quotient group $D^*/N(D_E^*)$, and more specifically how it behaves wrt base change.
Question 1: Let $Lsubseteq D$ be a maximal subfield of $D$. Is it true that $L^*/N(L_E^*)hookrightarrow D^*/N(D_E^*)$? That is, do we have $N(D_E)cap L=N(L_E)$? (Probably not...).
A positive answer to question 1 would give a some grip on the following question.
Question 2: consider an integer $n$ with the following property. Let $xin D$, and assume that for some big field $Ksupseteq F$, there exists $yin D_{Eotimes K}$ such that $N(y)=x$ in $D_K$. Then there exists a subfield $K_0subseteq K$ such that $x$ is a norm already over $K_0$ (i.e. there is $zin D_{Etimes K_0}$ such that $x=N(z)$ in $D_{K_0}$). What is the smallest $n$ with this property?
If this makes computations easier (though I don't think it will), it is OK to assume $p=3$.
abstract-algebra galois-theory division-algebras
asked 14 hours ago
Cehiju
364
Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago
|
show 3 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $E=F(u^{1/p})/F$ be a cyclic field extension of odd prime degree $p$, $D$ a cyclic central cyclic division algebra over $F$, and assume that $D_E$ is still a division algebra. We have a norm homomorphism $N:Eto F$, which induces another norm map $N:D_Eto D$.
I am trying to understand the quotient group $D^*/N(D_E^*)$, and more specifically how it behaves wrt base change.
Question 1: Let $Lsubseteq D$ be a maximal subfield of $D$. Is it true that $L^*/N(L_E^*)hookrightarrow D^*/N(D_E^*)$? That is, do we have $N(D_E)cap L=N(L_E)$? (Probably not...).
A positive answer to question 1 would give a some grip on the following question.
Question 2: consider an integer $n$ with the following property. Let $xin D$, and assume that for some big field $Ksupseteq F$, there exists $yin D_{Eotimes K}$ such that $N(y)=x$ in $D_K$. Then there exists a subfield $K_0subseteq K$ such that $x$ is a norm already over $K_0$ (i.e. there is $zin D_{Etimes K_0}$ such that $x=N(z)$ in $D_{K_0}$). What is the smallest $n$ with this property?
If this makes computations easier (though I don't think it will), it is OK to assume $p=3$.
abstract-algebra galois-theory division-algebras
asked 14 hours ago
Cehiju
364
Let $E=F(u^{1/p})/F$ be a cyclic field extension of odd prime degree $p$, $D$ a cyclic central cyclic division algebra over $F$, and assume that $D_E$ is still a division algebra. We have a norm homomorphism $N:Eto F$, which induces another norm map $N:D_Eto D$.
I am trying to understand the quotient group $D^*/N(D_E^*)$, and more specifically how it behaves wrt base change.
Question 1: Let $Lsubseteq D$ be a maximal subfield of $D$. Is it true that $L^*/N(L_E^*)hookrightarrow D^*/N(D_E^*)$? That is, do we have $N(D_E)cap L=N(L_E)$? (Probably not...).
A positive answer to question 1 would give a some grip on the following question.
Question 2: consider an integer $n$ with the following property. Let $xin D$, and assume that for some big field $Ksupseteq F$, there exists $yin D_{Eotimes K}$ such that $N(y)=x$ in $D_K$. Then there exists a subfield $K_0subseteq K$ such that $x$ is a norm already over $K_0$ (i.e. there is $zin D_{Etimes K_0}$ such that $x=N(z)$ in $D_{K_0}$). What is the smallest $n$ with this property?
If this makes computations easier (though I don't think it will), it is OK to assume $p=3$.
abstract-algebra galois-theory division-algebras
abstract-algebra galois-theory division-algebras
asked 14 hours ago
Cehiju
364
asked 14 hours ago
Cehiju
364
edited 8 hours ago
asked 14 hours ago
Cehiju
364
asked 14 hours ago
Cehiju
364
asked 14 hours ago
Cehiju
364
364
Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago
|
show 3 more comments
Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago
Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago
|
show 3 more comments
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Could you be more specific about the definition of the norm map $N:D_Eto D$, please. I'm not an expert, so I don't know how it goes. My go-to example of extending the rational quaternions to icosians amounts to extending $F=Bbb{Q}$ to $E=Bbb{Q}(sqrt5)$. Let $sigma$ be the non-trivial automorphism of $E$. It induces a mapping $sigma:D_Eto D_E$ by acting on the latter factor in the tensor product $D_E=Dotimes E$. I suspect the idea is to use a product of $id_{D_E}$ and $sigma$, but that fails to give elements of $D$.
– Jyrki Lahtonen
10 hours ago
For example, the "norm" $(i+sqrt5 j)(i-sqrt5 j)$ is not a rational quaternion.
– Jyrki Lahtonen
10 hours ago
Elements of $Dotimes E$ are sums of elements of the form $aotimes b$ where $ain D$ and $bin E$. Now let $N(aotimes b)=N(b)a$.
– Cehiju
8 hours ago
That can't be well-defined. $N(aotimes b)$ would then be different from $N(dfrac a2otimes 2b)$ even though $$aotimes b=frac{a}2otimes(2b).$$ Here $N(2b)=2^{[E:F]}N(b)$, so when $[E:F]>1$ we get different results.
– Jyrki Lahtonen
8 hours ago
Furthermore, how is that definition supposed to handle linear combinations like $a_1otimes b_1+a_2otimes b_2$?
– Jyrki Lahtonen
8 hours ago