Let (K, ||-||_K) be a normed field equipped with a non-Archimedean norm ||-||_K.

Proposition 7.1

(1) The set O={ x in K;  ||x||_K leq 1 }  is a commutative ring.

(2) m={ x in K; ||x||_K<1 } is the unique maximal ideal of O. (hence O is a local ring)

and U(K)={ x in K; ||x||_K=1 } is the group of units of O.

(3) K = Frac O, and O is integrally closed in K.

Proposition 7.2 

Sum_{k=1}^infty a_k, a_k in K, converges in K if and only if lim_{k --> +infty} a_k = 0.

Proposition 7.3 (Hensel's Lemma)

(Some sufficient conditions for existence of root of polynomial with coefficient in a complete field with non-Archimedean norm)

Let (K, ||-||_K) be a normed field equipped with a non-Archimedean norm ||-||_K. Let f(X) in O_K[X] and  a_0 in K be an element such that 

||f(a_0)||_K < || f'(a_0)||_K^2.

Then the sequence

a_{i+1}=a_i - f(a_i)/f'(a_i), i geq 0

converges in K to a root of f(X).

(We can't guarantee, in general, that Newton's approximation always converges. But Hensel's lemma tells us that it always converge for non-Archimedean case. So in some sense, non-Archimedean is easier.)

Proposition 7.4

Let f^bar(X）in k_K[X], non-zero, and let alpha^bar in k_K be a simple root of f^bar(X). Let f(X) in O_K[X] be such that f^bar(X) congruent f(X) mod m_K. Then there exists a unique root alpha in O_K of f(X) such that alpha^bar congruent alpha mod m_K.