Norm on k[X] or Q[x]?
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While considering specific examples of norms on number fields, I was considering $mathbb{Q}[sqrt{a}]cong frac{mathbb{Q}[x]}{(x^2-a)}$. This led me to the following question:
Given a field $k$, let $k[x]$ be the polynomial ring over $k$. Then the degree of $fin k[x]$ gives one a norm on $k[x]$. Does there exist another norm on $k[x]$ irrespective of the base field $k$?
I'm not sure how to find this even for $mathbb{Q}[x]$. I would appreciate any hints/references.
abstract-algebra number-theory norm polynomial-rings
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
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add a comment |
$begingroup$
While considering specific examples of norms on number fields, I was considering $mathbb{Q}[sqrt{a}]cong frac{mathbb{Q}[x]}{(x^2-a)}$. This led me to the following question:
Given a field $k$, let $k[x]$ be the polynomial ring over $k$. Then the degree of $fin k[x]$ gives one a norm on $k[x]$. Does there exist another norm on $k[x]$ irrespective of the base field $k$?
I'm not sure how to find this even for $mathbb{Q}[x]$. I would appreciate any hints/references.
abstract-algebra number-theory norm polynomial-rings
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
$endgroup$
2
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$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54
add a comment |
$begingroup$
While considering specific examples of norms on number fields, I was considering $mathbb{Q}[sqrt{a}]cong frac{mathbb{Q}[x]}{(x^2-a)}$. This led me to the following question:
Given a field $k$, let $k[x]$ be the polynomial ring over $k$. Then the degree of $fin k[x]$ gives one a norm on $k[x]$. Does there exist another norm on $k[x]$ irrespective of the base field $k$?
I'm not sure how to find this even for $mathbb{Q}[x]$. I would appreciate any hints/references.
abstract-algebra number-theory norm polynomial-rings
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
$endgroup$
While considering specific examples of norms on number fields, I was considering $mathbb{Q}[sqrt{a}]cong frac{mathbb{Q}[x]}{(x^2-a)}$. This led me to the following question:
Given a field $k$, let $k[x]$ be the polynomial ring over $k$. Then the degree of $fin k[x]$ gives one a norm on $k[x]$. Does there exist another norm on $k[x]$ irrespective of the base field $k$?
I'm not sure how to find this even for $mathbb{Q}[x]$. I would appreciate any hints/references.
abstract-algebra number-theory norm polynomial-rings
abstract-algebra number-theory norm polynomial-rings
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
asked Jan 8 at 0:43
Duttatrey Nath SrivastavaDuttatrey Nath Srivastava
93
93
2
$begingroup$
$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54
add a comment |
2
$begingroup$
$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54
2
2
$begingroup$
$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54
$begingroup$
$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54
add a comment |
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$begingroup$
$f mapsto q^{-deg(f)}$ is a non Archimedian absolute value on $k[x]$. For any number field $k$ let $sigma : k to mathbb{C}$ and $f^sigma(x)= sum_{n=0}^{deg(f)} sigma(a_n) x^n$ then $f mapsto |f^sigma(pi)|$ is an Archimedian absolute value on $k[x]$. Of course $pi$ can be replaced by any transcendental complex number. If $q >1 $ then the completion of $k(x), f mapsto q^{-deg(f)}$ is $k((x))$. The completion of $k(x), f mapsto |f^sigma(pi)|$ is $cong mathbb{R}$ or $mathbb{C}$.
$endgroup$
– reuns
Jan 8 at 0:54