Normed Fields

Definition 1: A seminorm on a ring R is a function p: R → R≥0 such that

(Nl) p(0) = 0;

(N2) p(-x) = p(x);

(N3) p(x + y) ≤ p(x) + p(y);

(N4) (submultiplicative) p(xy) ≤ p{x)p(y).

A seminorm p is called a norm on the ring R if it satisfies