He does assume they have already seen some non-archimedean fields, usually the p-adic numbers and hopefully the complete algebraically closed

p-adic field Cp, but he reviews them briefly just in case.

Infinite extension of the

p-adic field or the rational field, J.

Now, we can look for some applications to Hayman's problem in a p-adic field. Let f [member of] M (K).

Ojeda Hayman's Conjecture over a p-adic field. Taiwanese Journal of Mathematics.

Let X = [V.sub.P], the valuation ring of the

p-adic field [Q.sub.P].

They illustrate the theory with many examples, including matrix groups with entries in the field of real or complex numbers, or other locally compact fields such as

p-adic field, isometry groups of various metric spaces, and discrete groups themselves.

But the tools used in that study, such as properties of normal families, have no analogue on a

p-adic field. Here we shall use other methods, particularly the non-Archimedean Nevanlinna Theory.

In fact they established stability of Cauchy functional equations over

p-adic fields. In (17), (18) and (20) the stability of Cauchy, quadratic and quartic functional equations in non-Archimedean normed spaces were investigated.