Hensel's lemma: if f(a)≡0 (mod p) and f'(a)≢0 (mod p), then a lifts to a unique root mod p^k for all k. Formula: a₁ = a − f(a)·(f'(a))⁻¹ mod p². Like Newton's method but p-adically! Essential in algebraic number theory.
How to Use
Define polynomial and prime:
- Root mod p: Starting solution
- Lift: Solution mod p²,p³,...
- Steps: Each lifting step
Newton Analogy
Newton: x_{n+1} = x_n − f(x_n)/f'(x_n) converges in R. Hensel: a_{n+1} = a_n − f(a_n)·f'(a_n)⁻¹ converges in Q_p. Same formula, different 'distance'! Newton refines decimal digits; Hensel refines p-adic digits.
Applications
- Solving polynomial equations mod p^k
- Computing p-adic roots
- Algebraic number theory
- Constructing extensions of Q_p
Step-by-Step Instructions
- 1Enter polynomial.
- 2Enter prime p.
- 3Find root mod p.
- 4Apply Hensel lift.
- 5Get root mod p^k.