# , , and the ring of Witt vectors

No.05:

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

DEFINITION 05.1   A (unital commutative) ring is said to be a local ring if it has only one maximal ideal.

LEMMA 05.2   Let be a ring. Then the following conditions are equivalent:
1. is a local ring.
2. forms an ideal of .

PROPOSITION 05.3   is a local ring. Its maximal ideal is equal to .

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.

EXERCISE 05.1   Compute

EXERCISE 05.2   Find a solution to

such that .