Hensel's Lemma - Hensel's Lemma For p-adic Numbers

Hensel's Lemma For p-adic Numbers

In the p-adic numbers, where we can make sense of rational numbers modulo powers of p as long as the denominator is not a multiple of p, the recursion from r_k (roots mod pk) to r_k+1 (roots mod pk+1) can be expressed in a much more intuitive way. Instead of choosing t to be an(y) integer which solves the congruence, let t be the rational number (the pk here is not really a denominator since f(r_k) is divisible by pk). Then set

This fraction may not be an integer, but it is a p-adic integer, and the sequence of numbers r_k converges in the p-adic integers to a root of f(x) = 0. Moreover, the displayed recursive formula for the (new) number r_k+1 in terms of r_k is precisely Newton's method for finding roots to equations in the real numbers.

By working directly in the p-adics and using the p-adic absolute value, there is a version of Hensel's lemma which can be applied even if we start with a solution of f(a) ≡ 0 mod p such that f'(a) ≡ 0 mod p. We just need to make sure the number f'(a) is not exactly 0. This more general version is as follows: if there is an integer a which satisfies |f(a)|_p < |f′(a)|_p2, then there is a unique p-adic integer b such f(b) = 0 and |b-a|_p < |f'(a)|_p. The construction of b amounts to showing that the recursion from Newton's method with initial value a converges in the p-adics and we let b be the limit. The uniqueness of b as a root fitting the condition |b-a|_p < |f'(a)|_p needs additional work.

The statement of Hensel's lemma given above (taking ) is a special case of this more general version, since the conditions that f(a) ≡ 0 mod p and f'(a) ≠ 0 mod p say that |f(a)|_p < 1 and |f'(a)|_p = 1.