I know that $\mathbb{Z}[t,t^{-1}]$ is a localization of $\mathbb{Z}[t]$, the multiplicative set consisting of the non-negative powers of $t$.
But I do not know the maximal ideals of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$. What is the invertible element of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$?
Many thanks!
Note that $\mathbb Z[t,t^{-1}]\otimes \mathbb Q\cong \mathbb Q[t,t^{-1}]$ as this is just a special case of extension by scalars. Since $\mathbb Q[t,t^{-1}]\cong \mathbb Q[t]_{\{1,t,t^2,\ldots\}}$, its maximal ideals are in bijection with the maximal ideals of $\mathbb Q[t]$ which are disjoint from $\{1,t,t^2,\ldots\}$. Since $\mathbb Q[t]$ is a PID, the maximal ideals are $(p(t))$ for $p(t)$ a monic irreducible polynomial. This is disjoint from $\{1,t,t^2,\ldots\}$ unless $p(t)=t$, so the maximal ideals of $\mathbb Q[t,t^{-1}]$ are of the form $(p(t))$ with $p(t)\ne t$ an irreducible polynomial (WLOG with integer coefficients).
Under the isomorphism with $\mathbb Z[t,t^{-1}]$, these become $(p(t)\otimes 1)$ where $p(t)\ne t$ is irreducible, with positive leading coefficient and its coefficients have no common factor. By Gauss's lemma we do not need to distinguish between being irreducible over $\mathbb Z$ or over $\mathbb Q$.