Hensel's lemma

Hensel's lemma gives conditions under which a solution to a polynomial congruence modulo a prime can be lifted to compatible solutions modulo every higher prime power, defining a p-adic solution.

Newton's method: An iterative method that approximates a root by repeatedly using function values and derivatives. Hensel lifting is a p-adic analogue of Newton iteration, with divisibility controlling each correction.
Modular arithmetic: Arithmetic in which integers are identified when they have the same remainder modulo a fixed positive integer. The initial root and its successive lifts are solutions in residue rings.
p-adic analysis: The study of analysis over the p-adic numbers, using the p-adic absolute value. Hensel's lemma supplies roots and local inverse behavior needed in p-adic arguments.
Kurt Hensel: A German mathematician who introduced p-adic numbers and the lemma that bears his name. His work established the method of lifting roots through prime-power congruences.
p-adic valuation: The exponent of a prime p in a nonzero rational number, extended to p-adic numbers. Valuation measures how accurately each lifted approximation satisfies the polynomial equation.
Polynomial congruence: A statement that a polynomial evaluated at an integer is congruent to a specified value modulo an integer. Hensel's lemma begins with a polynomial congruence modulo p.
Local-global principle: A strategy that tests a global mathematical problem through its solutions over completions or local fields. Hensel lifting helps decide whether polynomial equations have solutions over p-adic fields.
p-adic numbers: A number system for each prime, built by completing the rationals with respect to divisibility by that prime. Hensel developed the lemma as a tool for working with these number systems.
Simple root: A root of a polynomial at which its derivative is nonzero. A nonzero derivative modulo p is the standard condition ensuring a unique lift.
Formal derivative: A derivative defined algebraically by applying the power rule to a polynomial. The polynomial's derivative determines whether a root can be lifted uniquely.

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