In this lecture, rings are assumed to be unital, associative and commutative unless otherwise specified.

- is a local ring.
- forms an ideal of .

We may do some ``analysis'' such as Newton's method to obtain some solution to algebraic equations.

Newton's method for approximating a solution of algebraic equation.

Let us solve an equation

in . We first note that

hold. So let us put as the first approximation of the solution. The Newton method tells us that for an approximation of the equation , a number calculated as

gives a better approximation.

So is a better approximation of the solution. In order to make the calculation easier, let us choose (insted of ) as a second approximation.

We choose as a second approximation.

We choose as a third approximation.

We choose as a third approximation.

such that .