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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 P

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 P

_{1}. 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.