^{1}

^{2, 3}

^{2, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

Two kinds of series representations, referred to as the Engel series and the Cohen-Egyptian fraction expansions, of elements in two different fields, namely, the real number and the discrete-valued non-archimedean fields are constructed. Both representations are shown to be identical in all cases except the case of real rational numbers.

It is well known [

Recall the following result, see, for example, Kapitel IV of [

Each

Define

We have

First, we show that

Concerning the rationality characterization, if its ES expansion is infinite periodic of period

In passing, we make the following observations.

For

If

If

To construct a Cohen-Egyptian fraction expansion, we proceed as in [

For any

Let

Each

To construct a CEF expansion for

To prove uniqueness, let

Finally, we look at its rationality characterization. If

The connection and distinction between ES and CEF expansions of a real number are described in the next theorem.

Let

(i) If

(ii) If

Both assertions follow mostly from Theorems

Let

In this case, we have

Thus,

We recapitulate some facts about discrete-valued non-archimedean fields taken from [

Every

Now we turn to the construction of a non-archimedean Cohen-Egyptian fraction expansion, in the same spirit as that of the real numbers, that is, by way of Lemma

For any

Let

For a non-archimedean CEF expansion, we now prove the following.

Each

Define

It is clear that the construction of non-archimedean ES and CEF expansions is identical which implies at once that the two representations are exactly the same in the non-archimedean case.

In the case of real numbers, we have seen that both ES and CEF expansions can be used to characterize rational numbers with quite different outcomes. In the non-archimedean situation, though ES and CEF expansions are identical, their use to characterize rational elements depend significantly on the underlying nature of each specific field. We end this paper by providing information on the rationality characterization in three different non-archimedean fields, namely, the field of

The following characterization of rational numbers by

Let

Now for function fields, we need more terminology. Let

For a characterization of rational elements, we prove the following.

The CEF of

Although the assertions in both fields

We use the notation of the last section with added subscripts

If the CEF of

Finally for the field

This work was supported by the Commission on Higher Education and the Thailand Research Fund RTA5180005 and by the Centre of Excellence In Mathematics, the Commission on Higher Education.