# Decomposition of the field of Laurent formal power series in a polynomial ring and a formal power series ring

Can we decompose the field of formal Laurent series as $$\mathbb C((t))\cong \mathbb C[t^{-1}]\times\mathbb C[[t]]$$ as vector spaces over the field of complex numbers? The map

$$\sum_{i\in\mathbb Z}\lambda_it^i\mapsto (\sum_{i<0}\lambda_it^i,\sum_{i\geq0}\lambda_it^i)$$

seems to satisfy the above decomposition.

• The RHS is not a field, so no such isomorphism exists Commented Jul 7, 2023 at 12:18
• Right, I agree with you on that. I made a sloppy mistake. I should have asked if the decomposition exists at the level of complex vector spaces, not algebras. I will edit the question. Commented Jul 7, 2023 at 12:21
• I think this map is not well-defined. What is the image of $\sum_{k\in \mathbb{Z}} t^k$ ? I believe you should replace $\mathbb{C}[t^{-1}]$ by $\mathbb{C}[[t^{-1}]]$. Commented Jul 7, 2023 at 12:25
• @groupoid formal Laurent series are typically defined to only contain series with finitely many terms of negative exponent. Commented Jul 7, 2023 at 12:28
• An element in $\mathbb C((t))$ has only finitely many terms $\lambda_k t^{k}$ with $k<0$, not infinitely many. So, the formal sum $\sum_{k\in\mathbb Z}t^k$ is not an element in the formal Laurent field, it seems. Commented Jul 7, 2023 at 12:33

• The isomorphism remains true if we replace the field $\mathbb C$ by a commutative ring $R$. Then, we simply have an isomorphism of $R$-modules, don't we? Commented Jul 7, 2023 at 12:39