In Alon's Nullstellensatz theorems (theorems 1.1 and 1.2 here) why is it necessary for $F$ to be a field? As far as I can tell, all the arguments in the proofs should work when $F$ is, say, an integral domain.
2026-03-26 03:00:53.1774494053
Why does Alon's combinatorial Nullstellensatz require working over a field.
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Here $ F $ is just simply a integral domain. You can use any domain. Like Ali commented if you take any part of $ F $ or any other integral domain then also the theorem will work.
Like example you have been asked to find the integer whose successor is 6. Many places you will have the equation $ x+1 = 6 $ but we know even if we take the equation as $ y+1=6$ then also nothing will be wrong.
Edit: by any part I mean substring