Number theory and Group theory

Similarly to Newton¢s Method, in p-adic analysis we find Hensel¢s Lemma, which allows to find the p-adic expansion of any given polynomial. Let f(x) be a polynomial function and we search its roots in the p-adic ring. First we need to define p. Then we find x₁ such that f(x₁) º 0 mod p. Then we look for x₂. We know that it is of the form x₂ = x₁ + pt and we need to determine t. We want that x₂ º x₁ modulo p. By the Taylor expansion we know that f(x₂) = f(x₁ + pt) = f(x₁) + ptf¢(x₁) +... We do not need to write all the expansion because f(x₁) + ptf¢(x₁) mod p² . Since we know that f(x₁) º 0 mod p we can divide the previous expression by p: f(x₁)/p + tf¢(x₁) º 0 mod p. We can find t from this expression and then we can find x₂ since it was defined as x₁ + pt. This steps can be done recursively similar to Newton¢s Method. In general, we use these two expressions: f(x_n-1)/p^n-1 + tf¢(x_n-1) º 0 mod p x_n = x_n-1 + tp^n-1 mod pⁿ . Also similar to Newton¢s Method, the necessary condition for Hensel¢s Lemma to work is that f¢(x) ¹ 0 mod p. Also, the coefficients of the polynomials need to be reduced modulo m. We can take as an example the equation x² = 5. We know that x₁ equals 4 and then x₂ = x₁ + pt = 4 + 11t mod p² . Then we find t by using the previous recursive expressions 11/11 + 8t º 0 mod p. We see that t equals 4. Then we know that x₂ = 4 + 11t = 4 + 11 · 4 = 48. Using this we obtain the different a_i which are 4, 4, 10, 4, 0...

Here you can see an example of the code for the polynomial x² -5.