Unramified extension of $L(sqrt{alpha})/L$
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I am studying an article of Chaoli and I try to understand the following statement:
If $L$ is a number field and $alpha in L^{times}/(L^{times})^{2}$ then, for an odd prime $p$, $L_{p}(sqrt{alpha})/L_{P}$ is unramified if and only if $alpha$ has even valuation.
for $p=2$, $L_{2}(sqrt{alpha})/L_{2}$ is unramified if and only if $alpha$ has even valuation and is represented by a $ unit equiv 1pmod{4}$
I try to use Kummer Theory but I don't know how can determine unramified extension of a local field?!
algebraic-number-theory extension-field
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
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add a comment |
$begingroup$
I am studying an article of Chaoli and I try to understand the following statement:
If $L$ is a number field and $alpha in L^{times}/(L^{times})^{2}$ then, for an odd prime $p$, $L_{p}(sqrt{alpha})/L_{P}$ is unramified if and only if $alpha$ has even valuation.
for $p=2$, $L_{2}(sqrt{alpha})/L_{2}$ is unramified if and only if $alpha$ has even valuation and is represented by a $ unit equiv 1pmod{4}$
I try to use Kummer Theory but I don't know how can determine unramified extension of a local field?!
algebraic-number-theory extension-field
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
$endgroup$
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See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
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– MiRi_NaE
Jan 14 at 9:44
1
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Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21
add a comment |
$begingroup$
I am studying an article of Chaoli and I try to understand the following statement:
If $L$ is a number field and $alpha in L^{times}/(L^{times})^{2}$ then, for an odd prime $p$, $L_{p}(sqrt{alpha})/L_{P}$ is unramified if and only if $alpha$ has even valuation.
for $p=2$, $L_{2}(sqrt{alpha})/L_{2}$ is unramified if and only if $alpha$ has even valuation and is represented by a $ unit equiv 1pmod{4}$
I try to use Kummer Theory but I don't know how can determine unramified extension of a local field?!
algebraic-number-theory extension-field
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
$endgroup$
I am studying an article of Chaoli and I try to understand the following statement:
If $L$ is a number field and $alpha in L^{times}/(L^{times})^{2}$ then, for an odd prime $p$, $L_{p}(sqrt{alpha})/L_{P}$ is unramified if and only if $alpha$ has even valuation.
for $p=2$, $L_{2}(sqrt{alpha})/L_{2}$ is unramified if and only if $alpha$ has even valuation and is represented by a $ unit equiv 1pmod{4}$
I try to use Kummer Theory but I don't know how can determine unramified extension of a local field?!
algebraic-number-theory extension-field
algebraic-number-theory extension-field
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
asked Jan 12 at 20:10
Mohammad BabakhaniMohammad Babakhani
667
667
$begingroup$
See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
$endgroup$
– MiRi_NaE
Jan 14 at 9:44
1
$begingroup$
Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21
add a comment |
$begingroup$
See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
$endgroup$
– MiRi_NaE
Jan 14 at 9:44
1
$begingroup$
Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21
$begingroup$
See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
$endgroup$
– MiRi_NaE
Jan 14 at 9:44
$begingroup$
See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
$endgroup$
– MiRi_NaE
Jan 14 at 9:44
1
1
$begingroup$
Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21
$begingroup$
Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21
add a comment |
1 Answer
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The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $mu_p$ of $p$-th roots of unity, and let $L=K(sqrt [p]alpha)$ for some $alpha in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $vnmid p $: The place $v$ is unramified in $L/K$ iff $v(alpha) equiv 0$ mod $p$ ; (ii) Wild case, $vmid p$: Let $e_v$ be the ramification index of $v$ in $K/mathbf Q(mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v in K^*$ s.t. $(alpha/{x_v}^p)equiv 1$ mod ${frak P_v}^{pe_v}$.
Idea of proof : (i) is clear. For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
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$begingroup$
The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $mu_p$ of $p$-th roots of unity, and let $L=K(sqrt [p]alpha)$ for some $alpha in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $vnmid p $: The place $v$ is unramified in $L/K$ iff $v(alpha) equiv 0$ mod $p$ ; (ii) Wild case, $vmid p$: Let $e_v$ be the ramification index of $v$ in $K/mathbf Q(mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v in K^*$ s.t. $(alpha/{x_v}^p)equiv 1$ mod ${frak P_v}^{pe_v}$.
Idea of proof : (i) is clear. For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
$endgroup$
add a comment |
$begingroup$
The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $mu_p$ of $p$-th roots of unity, and let $L=K(sqrt [p]alpha)$ for some $alpha in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $vnmid p $: The place $v$ is unramified in $L/K$ iff $v(alpha) equiv 0$ mod $p$ ; (ii) Wild case, $vmid p$: Let $e_v$ be the ramification index of $v$ in $K/mathbf Q(mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v in K^*$ s.t. $(alpha/{x_v}^p)equiv 1$ mod ${frak P_v}^{pe_v}$.
Idea of proof : (i) is clear. For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
$endgroup$
add a comment |
$begingroup$
The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $mu_p$ of $p$-th roots of unity, and let $L=K(sqrt [p]alpha)$ for some $alpha in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $vnmid p $: The place $v$ is unramified in $L/K$ iff $v(alpha) equiv 0$ mod $p$ ; (ii) Wild case, $vmid p$: Let $e_v$ be the ramification index of $v$ in $K/mathbf Q(mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v in K^*$ s.t. $(alpha/{x_v}^p)equiv 1$ mod ${frak P_v}^{pe_v}$.
Idea of proof : (i) is clear. For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
$endgroup$
The ramification results obtained from Kummer theory are classically known and are rather complete, at least in the case of a kummerian extension of prime degree $p$. See e.g. G. Gras' book, "CFT-From theory to practice" (Springer),chap. I,§6. Here is a summary centered on your question (the notations are those of the book):
Let $K$ be a number field containing the group $mu_p$ of $p$-th roots of unity, and let $L=K(sqrt [p]alpha)$ for some $alpha in K^*/{K^*}^p$. Let $v$ be a finite place of $K$. Then : (i) Tame case, $vnmid p $: The place $v$ is unramified in $L/K$ iff $v(alpha) equiv 0$ mod $p$ ; (ii) Wild case, $vmid p$: Let $e_v$ be the ramification index of $v$ in $K/mathbf Q(mu_p)$. Then $v$ is unramified in $L/K$ iff there exists $x_v in K^*$ s.t. $(alpha/{x_v}^p)equiv 1$ mod ${frak P_v}^{pe_v}$.
Idea of proof : (i) is clear. For (ii), it is well known that $K_v$ has a unique unramified cyclic extension of degree $p$, obtained from $K_v$ by adding a root of unity of order prime to $p$. The technical point here is to express this extension as a kummerian extension (loc. cit.)
You can apply this to your case with $p=2$.
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
answered Jan 14 at 10:54
nguyen quang donguyen quang do
8,9991724
8,9991724
add a comment |
add a comment |
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See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
$endgroup$
– MiRi_NaE
Jan 14 at 9:44
1
$begingroup$
Mohammad. You cannot @-ping users who haven't participated in this thread. Some may feel that such pings amount to harassment. My own feelings would vary, depending.
$endgroup$
– Jyrki Lahtonen
Jan 14 at 11:21