Newton's method
Over p -adic numbers

In p -adic analysis, the standard method to show a polynomial equation in one variable has a p -adic root is Hensel's lemma, which uses the recursion from Newton's method on the p -adic numbers. Because of the more stable behavior of addition and multiplication in the p -adic numbers compared to the real numbers (specifically, the unit ball in the p -adics is a ring), convergence in Hensel's lemma can be guaranteed under much simpler hypotheses than in the classical Newton's method on the real line.
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