ℤ_{p}, ℚ_{p}, AND THE RING OF WITT VECTORS

No.05: ℤ_{p} as a local ring. In this lecture, rings are assumed to be unital, associative and com-

mutative unless otherwise specified.

Definition 5.1. A (unital commutative) ring A is said to be a local ring if it has only one maximal ideal.

Lemma 5.2. Let A be a ring. Then the following conditions are equivalent:

(1) A is a local ring.

(2) A \ A^{×} forms an ideal of A.

Proposition 5.3. ℤ_{p} is a local ring. Its maximal ideal is equal to pℤ_{p}.

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

x^{2} = 2

in ℤ_{7}. We first note that

3^{2} ≡ 2 (7)

hold. So let us put x_{0} = 3 = [0.3]_{7} as the first approximation of the solution. The Newton method tells us that for an approximation x of the equation x^{2} = 2, a number x′ calculated as

x′ = \frac{1}{2}(x + \frac{2}{x})

gives a better approximation.

x′_{0} = \frac{1}{2}([0.3]_{7} + [0.3˙2]_{7} = [0.3˙1]_{7}

So [0.3˙1]_{7} is a better approximation of the solution. In order to make the calculation easier, let us choose x_{1} = [0.31]_{7} (insted of x′_{0}) as a second approximation.

x′_{1} = \frac{1}{2}([0.31]_{7} + 2/[0.31]_{7}) = \frac{1}{2}([0.31]_{7} + [0.3˙145˙2]_{7}) ≒ [0.312]_{7}

We choose x_{2} = [0.312]_{7} as a second approximation.

x′_{2} = \frac{1}{2}([0.312]_{7} + 2/[0.3˙12534066˙2]_{7}) ≒ [0.31261]_{7}

We choose x_{3} = [0.31261]_{7} as a third approximation.

x′_{3} == \frac{1}{2}([0.31261]_{7} + [0.3126142465066 …]_{7}) ≒ [0.312612124...]_{7}

We choose x_{4} = [0.312612124]_{7} as a third approximation.

x′_{4} = \frac{1}{2}([0.312612124]_{7} + [0.312612124565220422662213135351 …]_{7})

≒ [0.3126121246621102]_{7}

Exercise 5.1. Compute [0.5]_{7}/[0.11]_{7}

Exercise 5.2. Find a solution to

x^{3} ≡ 5 (mod 11^{5})

such that x ≡ 3 (mod 11).