Mixing
Class field axiom III-1 in Serre's “Local Fields”
$begingroup$
Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* to E^*$ be the norm map (it is continuos).
And consider an axiom:
$N$ has closed image and compact kernel.
In the book, the author says that this condition is equivalent to that $N$ is proper.
But I can't show it.
Please help.
P.S. This condition holds in this case.
I use the word "axiom" because I use the explicit terminology for simplification (the author uses class field formation).
So please show the equivalence using only locally compactness, countability at infinity of $E, F$ and continuity of $N$.
algebraic-number-theory class-field-theory
edited Feb 22 at 0:22
user38077
31
asked Jan 30 at 8:53
agababibuagababibu
413110
$endgroup$
add a comment |
$begingroup$
Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* to E^*$ be the norm map (it is continuos).
And consider an axiom:
$N$ has closed image and compact kernel.
In the book, the author says that this condition is equivalent to that $N$ is proper.
But I can't show it.
Please help.
P.S. This condition holds in this case.
I use the word "axiom" because I use the explicit terminology for simplification (the author uses class field formation).
So please show the equivalence using only locally compactness, countability at infinity of $E, F$ and continuity of $N$.
algebraic-number-theory class-field-theory
edited Feb 22 at 0:22
user38077
31
asked Jan 30 at 8:53
agababibuagababibu
413110
$endgroup$
add a comment |
$begingroup$
Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* to E^*$ be the norm map (it is continuos).
And consider an axiom:
$N$ has closed image and compact kernel.
In the book, the author says that this condition is equivalent to that $N$ is proper.
But I can't show it.
Please help.
P.S. This condition holds in this case.
I use the word "axiom" because I use the explicit terminology for simplification (the author uses class field formation).
So please show the equivalence using only locally compactness, countability at infinity of $E, F$ and continuity of $N$.
algebraic-number-theory class-field-theory
edited Feb 22 at 0:22
user38077
31
asked Jan 30 at 8:53
agababibuagababibu
413110
$endgroup$
Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* to E^*$ be the norm map (it is continuos).
And consider an axiom:
$N$ has closed image and compact kernel.
In the book, the author says that this condition is equivalent to that $N$ is proper.
But I can't show it.
Please help.
P.S. This condition holds in this case.
I use the word "axiom" because I use the explicit terminology for simplification (the author uses class field formation).
So please show the equivalence using only locally compactness, countability at infinity of $E, F$ and continuity of $N$.
algebraic-number-theory class-field-theory
algebraic-number-theory class-field-theory
edited Feb 22 at 0:22
user38077
31
asked Jan 30 at 8:53
agababibuagababibu
413110
edited Feb 22 at 0:22
user38077
31
asked Jan 30 at 8:53
agababibuagababibu
413110
edited Feb 22 at 0:22
user38077
31
edited Feb 22 at 0:22
user38077
31
edited Feb 22 at 0:22
user38077
31
31
asked Jan 30 at 8:53
agababibuagababibu
413110
asked Jan 30 at 8:53
agababibuagababibu
413110
asked Jan 30 at 8:53
agababibuagababibu
413110
413110
add a comment |
add a comment |
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