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Algebraic closure of $mathbbQ$ in $mathbbQ_p$

2

1

$begingroup$

Let $(K, |cdot |)$ be a (discrete) valuation field, and $(widehatK, |cdot|)$ be its completion. Then we can think about the algebraic (or separable) closure of $K$ in $widehatK$, which is called Henselization of $K$.

I want to know how to describe the Henselization of a given field. For example. Let's assume that we have a fixed (odd?) prime $p$ and $K = mathbbQ$ with a $p$-adic norm $|cdot |_p$.
What is the Henselization, i.e. algebraic closure of $mathbbQ$ in $mathbbQ_p$?
This is strictly smaller than $mathbbQ_p$ since not every element in $mathbbQ_p$ is algebraic over $mathbbQ$. Also, we can find some nontrivial examples which are in the Henselization. For example, if $p = 5$ then $sqrt11$ is in $mathbbQ_5$ and so in the Henselization.
Is there an explicit way to describe elements in that field?
Do we have some set-theoretical issue here?

p-adic-number-theory

share|cite|improve this question

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why would there be any set-theoretic issues?
  $endgroup$
  – Qiaochu Yuan
  Apr 8 at 21:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @QiaochuYuan Actually it is not an issue, but I just want to talk about something like Hamel basis, which we know the existence (assuming AC) but there's no way to construct it explicitly.
  $endgroup$
  – Seewoo Lee
  Apr 8 at 23:23
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I don't really understand what would count as a good description for you, here. Are you satisfied with the description of $overlinemathbbQ$ as the algebraic closure of $mathbbQ$, despite e.g. not being able to write down a Hamel basis of it? Would you be satisfied with knowing which polynomials over $mathbbQ$ have roots in $mathbbQ_p$? That shouldn't be hard to settle using Hensel's lemma (which I think is why this is called the Henselization).
  $endgroup$
  – Qiaochu Yuan
  Apr 9 at 3:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Just a few days ago there was this question with the exact same title: math.stackexchange.com/q/3172907/96384. While I cannot really advertise my near-trivial answer there, the comments contain a link to this question and its answers, and linked from there is this helpful MathOverflow post: mathoverflow.net/q/17032/27465.
  $endgroup$
  – Torsten Schoeneberg
  Apr 9 at 4:32
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @QiaochuYuan Now I understand that I asked a stupid question.
  $endgroup$
  – Seewoo Lee
  Apr 10 at 0:32
  

  
  
  
  

 | 
show 1 more comment

2

1

$begingroup$

Let $(K, |cdot |)$ be a (discrete) valuation field, and $(widehatK, |cdot|)$ be its completion. Then we can think about the algebraic (or separable) closure of $K$ in $widehatK$, which is called Henselization of $K$.

I want to know how to describe the Henselization of a given field. For example. Let's assume that we have a fixed (odd?) prime $p$ and $K = mathbbQ$ with a $p$-adic norm $|cdot |_p$.
What is the Henselization, i.e. algebraic closure of $mathbbQ$ in $mathbbQ_p$?
This is strictly smaller than $mathbbQ_p$ since not every element in $mathbbQ_p$ is algebraic over $mathbbQ$. Also, we can find some nontrivial examples which are in the Henselization. For example, if $p = 5$ then $sqrt11$ is in $mathbbQ_5$ and so in the Henselization.
Is there an explicit way to describe elements in that field?
Do we have some set-theoretical issue here?

p-adic-number-theory

share|cite|improve this question

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Why would there be any set-theoretic issues?
  $endgroup$
  – Qiaochu Yuan
  Apr 8 at 21:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @QiaochuYuan Actually it is not an issue, but I just want to talk about something like Hamel basis, which we know the existence (assuming AC) but there's no way to construct it explicitly.
  $endgroup$
  – Seewoo Lee
  Apr 8 at 23:23
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  I don't really understand what would count as a good description for you, here. Are you satisfied with the description of $overlinemathbbQ$ as the algebraic closure of $mathbbQ$, despite e.g. not being able to write down a Hamel basis of it? Would you be satisfied with knowing which polynomials over $mathbbQ$ have roots in $mathbbQ_p$? That shouldn't be hard to settle using Hensel's lemma (which I think is why this is called the Henselization).
  $endgroup$
  – Qiaochu Yuan
  Apr 9 at 3:57
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Just a few days ago there was this question with the exact same title: math.stackexchange.com/q/3172907/96384. While I cannot really advertise my near-trivial answer there, the comments contain a link to this question and its answers, and linked from there is this helpful MathOverflow post: mathoverflow.net/q/17032/27465.
  $endgroup$
  – Torsten Schoeneberg
  Apr 9 at 4:32
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @QiaochuYuan Now I understand that I asked a stupid question.
  $endgroup$
  – Seewoo Lee
  Apr 10 at 0:32
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Let $(K, |cdot |)$ be a (discrete) valuation field, and $(widehatK, |cdot|)$ be its completion. Then we can think about the algebraic (or separable) closure of $K$ in $widehatK$, which is called Henselization of $K$.

I want to know how to describe the Henselization of a given field. For example. Let's assume that we have a fixed (odd?) prime $p$ and $K = mathbbQ$ with a $p$-adic norm $|cdot |_p$.
What is the Henselization, i.e. algebraic closure of $mathbbQ$ in $mathbbQ_p$?
This is strictly smaller than $mathbbQ_p$ since not every element in $mathbbQ_p$ is algebraic over $mathbbQ$. Also, we can find some nontrivial examples which are in the Henselization. For example, if $p = 5$ then $sqrt11$ is in $mathbbQ_5$ and so in the Henselization.
Is there an explicit way to describe elements in that field?
Do we have some set-theoretical issue here?

p-adic-number-theory

share|cite|improve this question

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

$endgroup$

share|cite|improve this question

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

share|cite|improve this question

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930

asked Apr 8 at 20:28

Seewoo LeeSeewoo Lee

7,2542930