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cleaned up a lot of bad English; deleted 1 characters in body; edited title
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KConrad
KConrad
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local class field theory (Norm operatormap)

Let $K$ be a local field, for example the $p$-adic numbers. In the standardNeukirch's book: Algebraic "Algebraic number theory, written by Jurgen Neukirchtheory", there is onethe statement:

If if $K$ contains the $n$-th roots of unity and if the charactericcharacteristic of $K$ does not divide $n$, and we consider the extensionset $L=L(\sqrt[n]{K^{\times}})$ over K$L=K(\sqrt[n]{K^{\times}})$, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are the following:

Assume L/F is galois field, for a (finite) Galois extension $L$ of $K$:\

(1) What happens if charactericthe characteristic of $K$ dividedivides $n$ , can? Can one obtain thean explicit form of the image of the norm  of $L^\times$?\

(2) If we don't add the condition: that $K$ contains the $n$-th roots of unity, what is the image of the norm operator, it may be relative to? If $K^{\times n}$$L = K(x)$, can one write it in terms of the generatorprimitive element $x$ and $K^{\times n}$ if we know L=F(x) ?\

local class field theory (Norm operator)

Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:

If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\times}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are following:

Assume L/F is galois field extension :

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm  ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\

local class field theory (Norm map)

Let $K$ be a local field, for example the $p$-adic numbers. In Neukirch's book "Algebraic number theory", there is the statement: if $K$ contains the $n$-th roots of unity and if the characteristic of $K$ does not divide $n$, and we set $L=K(\sqrt[n]{K^{\times}})$, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are the following, for a (finite) Galois extension $L$ of $K$:\

(1) What happens if the characteristic of $K$ divides $n$? Can one obtain an explicit form of the image of the norm of $L^\times$?\

(2) If we don't add the condition that $K$ contains the $n$-th roots of unity, what is the image of the norm operator? If $L = K(x)$, can one write it in terms of the primitive element $x$ and $K^{\times n}$?\

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user13726
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Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:\ If

If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\start}})$$L=L(\sqrt[n]{K^{\times}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.\ My

My questions are Assume following:

Assume L/F is galois field extension :\

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\

Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:\ If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\start}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.\ My questions are Assume L/F is galois field extension :\

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\

Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:

If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\times}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.

My questions are following:

Assume L/F is galois field extension :

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\

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user13726
user13726
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local class field theory (Norm operator)

Let $K$ be a local field. In the standard book: Algebraic number theory, written by Jurgen Neukirch, there is one statement:\ If $K$ contains the $n$-th roots of unity and if the characteric of $K$ does not divide $n$, we consider the extension $L=L(\sqrt[n]{K^{\start}})$ over K, then one has $N_{L/K}(L^{\times})=K^{\times n}$.\ My questions are Assume L/F is galois field extension :\

(1) What happens if characteric of $K$ divide $n$ , can one obtain the explicit form of the image of the norm ?\

(2) If we don't add the condition: $K$ contains $n$-th roots, what is the image of the norm operator, it may be relative to $K^{\times n}$, can one write it in terms of the generator $x$ and $K^{\times n}$ if we know L=F(x) ?\