Ax-Kochen theorem
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The Ax-Kochen theorem, named for James Ax and Simon Kochen, states that for any given natural number d every homogeneous polynomial of degree d over the p-adic numbers in at least d2+1 variables represents 0 for all but finitely many prime numbers p.James Ax and Simon Kochen, Diophantine problems over local fields I., American Journal of Mathematics, 87, pages 605-630, (1965)
The proof of the theorem makes extensive use of methods from mathematical logic, such as model theory.
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