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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
   • (edited Nov 20th 2011)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI saw a question on MO that intrigued me. If one has a polynomial of one variable and looks at the symmetries you get the Galois group of the corresponding extension. What happens if you take a polynomial in two variables? The inclusion of the field k into k[x,y]/ f(x,y) induces a map of schemes and my instinct is to look for the object of symmetries of that, but it should be higher-dimensional, hence it might be more like a 2-group than a group. I may be very wrong on this, but wondered if anyone who was in on the discussions on 2-geometry has seen anything like that. It seems to be to be like a fibration with some curves as fibres. Any thought would be very welcome. :-)

   I saw a question on MO that intrigued me. If one has a polynomial of one variable and looks at the symmetries you get the Galois group of the corresponding extension. What happens if you take a polynomial in two variables? The inclusion of the field k into k[x,y]/ f(x,y) induces a map of schemes and my instinct is to look for the object of symmetries of that, but it should be higher-dimensional, hence it might be more like a 2-group than a group. I may be very wrong on this, but wondered if anyone who was in on the discussions on 2-geometry has seen anything like that. It seems to be to be like a fibration with some curves as fibres. Any thought would be very welcome. :-)

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 20th 2011
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   Author: David_Corfield
   Format: MarkdownItexMinhyong Kim has some ideas on a Galois theory for two variables. Follow the link to the talk from [here](http://golem.ph.utexas.edu/category/2009/12/galois_theory_in_two_variables.html).

   Minhyong Kim has some ideas on a Galois theory for two variables. Follow the link to the talk from here.

3. • CommentRowNumber3.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
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   Author: Tim_Porter
   Format: MarkdownItexWill do, Ta.

   Will do, Ta.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 20th 2011
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   Author: David_Corfield
   Format: MarkdownItexI see one can't drag the cursor to the end of the talk. Now I'm wondering how much Minhyong says about two variable Galois theory. Perhaps it's only just briefly mentioned. (I was there, but it was a while ago.)

   I see one can’t drag the cursor to the end of the talk. Now I’m wondering how much Minhyong says about two variable Galois theory. Perhaps it’s only just briefly mentioned. (I was there, but it was a while ago.)

5. • CommentRowNumber5.
   • CommentAuthorTim_Porter
   • CommentTimeNov 20th 2011
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexThe cursor drag works well for me????? The two variable Galois theory is lurking behind all this I feel. I not he uses conjugation action several times and that always hides a homotopy , but in the particular cases that I undestood, I could not see the crossed module behind the construction. but it may be that it is a groupoid on which the crossed module is acting or something like that. That begins to feel 2-torsorish. ;-)

   The cursor drag works well for me????? The two variable Galois theory is lurking behind all this I feel. I not he uses conjugation action several times and that always hides a homotopy , but in the particular cases that I undestood, I could not see the crossed module behind the construction. but it may be that it is a groupoid on which the crossed module is acting or something like that. That begins to feel 2-torsorish. ;-)