Valuation ring - can someone explain this ?

  • #1
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Main Question or Discussion Point

Hello,
I asked somebody a question, and didn't understood his answer. Can someone explain it to me ?
My question was : Is there a valuation ring in ℚ(x,y), lying above the ideal <x,y> in the ring ℚ[x,y], whose residual field is a non-trivial extension of ℚ ? Here is his answer:
This is not too hard, geometrically blow up the origin, then there are infinitely many points on that P1. If you blowup any one of those points, that gives a distinct discrete valuation lying over the origin. The residue field of every such valuation ring is a transcendental (degree = 1) extension of ℚ.
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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