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?!










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  • $begingroup$
    See Neukirch's `Algebraic Number Theory', Lemma 3.3, Ch.5 for odd $p$.
    $endgroup$
    – 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
















2












$begingroup$


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?!










share|cite|improve this question









$endgroup$












  • $begingroup$
    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














2












2








2


1



$begingroup$


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?!










share|cite|improve this question









$endgroup$




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






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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


















  • $begingroup$
    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










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 $p=2$.






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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 $p=2$.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      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 $p=2$.






      share|cite|improve this answer









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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 $p=2$.






        share|cite|improve this answer









        $endgroup$



        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 $p=2$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 14 at 10:54









        nguyen quang donguyen quang do

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        8,9991724






























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