Field and embedded map

Question

Let $S=\{a\in \mathbb{R}^{\mathbb{Q}}| (\exists k\in\mathbb{Z})(\forall n<k)a_n=0\}$ and let $+$ and $\cdot$ be binary operations defined with: $$(a+b)_n=a_n+b_n,\quad (a\cdot b)_n=\sum_{i+j=n}a_i,b_j,\qquad a,b\in\mathbb{R}^{\mathbb{Q}}, n\in\mathbb{Z}.$$ Constants $\mathbf{0},\mathbf{1},\varepsilon$ we define with $\mathbf{0}=\langle 0 | n\in\mathbb{Z}\rangle,$ $\mathbf{1}_n=1$ if $n=0,$ otherwise $\mathbf{1}_n=0,$ and $\varepsilon_n=1$ if $n=1,$ otherwise $\varepsilon_n=0,$ $n\in\mathbb{N}.$ How to prove that $\mathbf{S}=(S,+,\cdot,\mathbb{0},\mathbb{1})$ is a field and how to show that there exists embedded map $h:\mathbb{R}\rightarrow \mathbf{S}?$

Answer 1

(If you really meant $n\in \Bbb{Q}$) then you forgot an important condition: that for any $N$ there are only finitely many $n<N$ such that $a_n\ne 0$.

Your ring is then the ring of formal series $$\Bbb{R}((x^\Bbb{Q}))=\{ \sum_{j\ge 0} c_j x^{r_j}, c_j\in \Bbb{R},r_j\in \Bbb{Q}, \lim_{j\to \infty} r_j=\infty\}$$

It is obviously an integral domain. If $r_0=0,c_0=1$ and all the others $r_j > 0$ then the inverse is $$1+ \sum_{k\ge 1}(- \sum_{j\ge 1} c_j x^{r_j})^k$$ which proves that it is a field.

Note that $\overline{\Bbb{Q}}((x^\Bbb{Q}))$ is algebraically closed and complete, it is abstractly isomorphic to $\Bbb{C}$ but with an alternative topology to that of $\Bbb{C}$ and $\Bbb{C}_p$.