## what is the structure of $R \otimes \mathbb{Q}[[X]]$?

Let $$R$$ be a ring and $$\mathbb{Q}[[X]]$$ be the ring formal power series in rational field $$\mathbb{Q}$$. Let $$f(X) \in R \otimes \mathbb{Q}[[X]]$$ be a power series in $$X$$.

My question is-

How does $$R \otimes \mathbb{Q}[[X]]$$ look like or what is the structure of $$R \otimes \mathbb{Q}[[X]]$$ ?

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## Suppose $\zeta$ is a root of unity and $f\in \mathbb{Q}[X]$. Show that $f(\zeta)\neq2^{1/4}$.

Question: Suppose $$\zeta$$ is a root of unity and $$f\in \mathbb{Q}[X]$$. Show that $$f(\zeta)\neq2^{1/4}$$.

This is from the UCLA fall ’16 algebra qual. So far I haven’t gotten far besides my initial observation that if we suppose that $$f(\zeta)=2^{1/4}$$, then as $$\mathbb{Q}(\zeta)/\mathbb{Q}$$ is Galois, we get $$g(X)=X^4-2$$ splits in $$\mathbb{Q}$$ and so we have $$\mathbb{Q}(\zeta)/E/\mathbb{Q}$$ where $$E$$ is the splitting field of $$g$$. Since $$|E:\mathbb{Q}|=2^3$$ we get that $$2^3|\varphi(ord(\zeta))$$ and this eliminates a lot of possible roots of unities. However, proving the general result escapes me.

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