every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.
1
$begingroup$
Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $mathbb{Z}$.
How to prove that every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.
valuation-theory ordered-fields
asked Dec 2 '18 at 13:02
XYZXYZ
978
$endgroup$
|
show 8 more comments
1
$begingroup$
Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $mathbb{Z}$.
How to prove that every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.
valuation-theory ordered-fields
asked Dec 2 '18 at 13:02
XYZXYZ
978
$endgroup$
$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29
|
show 8 more comments
1
1
1
$begingroup$
Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $mathbb{Z}$.
How to prove that every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.
valuation-theory ordered-fields
asked Dec 2 '18 at 13:02
XYZXYZ
978
$endgroup$
Natural valuation has a convex valuation ring, in fact the valuation ring is the convex hull of $mathbb{Z}$.
How to prove that every ordered field $K$ has a natural valuation $v$, whose residue field is an archimedean ordered field.
valuation-theory ordered-fields
valuation-theory ordered-fields
asked Dec 2 '18 at 13:02
XYZXYZ
978
asked Dec 2 '18 at 13:02
XYZXYZ
978
asked Dec 2 '18 at 13:02
XYZXYZ
978
asked Dec 2 '18 at 13:02
XYZXYZ
978
asked Dec 2 '18 at 13:02
XYZXYZ
978
978
$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29
|
show 8 more comments
$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29
$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29
|
show 8 more comments
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$begingroup$
Well you can first try to prove that the convex hull of $mathbb{Z}$ in $K$ is a valuation ring of $K$.
$endgroup$
– nombre
Dec 2 '18 at 13:40
$begingroup$
There is a theorem that gives that result right away. Convex hull of any subring in ordred field K is valuation ring. Therefore this one too.
$endgroup$
– XYZ
Dec 2 '18 at 14:03
$begingroup$
Okay, then do you know how the order on this residue field is defined? If so, given $x in operatorname{Hull}(mathbb{Z})$, can you find $n in mathbb{N}$ with $overline{x} leq overline{n}$? (where $overline{a}= a+mathfrak{m}$ and $mathfrak{m}$ is the maximal ideal of the valuation ring)
$endgroup$
– nombre
Dec 2 '18 at 15:28
$begingroup$
I know that $P={bar a : ain mathbb{Z} , ageq 0 }$ is an ordering in residue field. How to use it to find $n$?
$endgroup$
– XYZ
Dec 2 '18 at 16:18
$begingroup$
Correction: $a$ in ordering P is in convex hull of $mathbb{Z}$
$endgroup$
– XYZ
Dec 2 '18 at 16:29