Field of quotients

Construct the field of fractions $\mathbb {R}((X))$ of the ring $\mathbb {R}{}[[X]]$ of formal power series in $X$ with coefficients in the field $\mathbb {R}$. Relying on Exercise 6 from $\S 3$, show that every element of the field $\mathbb {R}((X))$ has the form of the so-called meromorphic power series

in which a finite number of negative exponents is allowed. In other words, $\varphi (X)=X^{-m} f(X)$, where $f(X)$ is an ordinary power series from $\mathbb {R}{}[[X]]$.

Let an infinite sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ be periodic, starting from some term. Show that the power series $f(X)=a_{0}+a_{1} X+a_{2} X^{2}+\ldots$ can be written as a rational fraction from $\mathbb {R}(X)$.

Show that the set of simplest fractions of the form $f / p^{n}, n \geqslant 1$ (and their linear combinations) is a subring of the ring $P_{0}(X)$ (see the remark at the end of item 2).

Show that, in accordance with Exercise 3, the simplest fractions over $P=\mathbb {Z}_{3}$

form a ring $K$ with an infinitely descending chain

of subrings $K_{N}$ spanned by the fractions $(a X+b) /\left(X^{2}+1\right)^{n}, n \geqslant N$.