This is something I was curious about, but not really sure why. I was hoping maybe someone could give me a good explanation. I understand that there are similarities between $\mathbb{Z}$ and $\mathbb{F}_q[t]$, where $\mathbb{F}_q$ is a finite field of $q$ elements. And many interesting results in number theory over $\mathbb{Z}$ have an $\mathbb{F}_q[t]$-analogue. When we have some result over $\mathbb{Z}$ (or $\mathbb{Q}$), and we consider these analogues, do we generalize to $\mathbb{F}_q[t]$ (or $\mathbb{F}_q(t)$) just for the sake of generalizing it to a different setting? or are there more reasons why this generalization to $\mathbb{F}_q[t]$ is interesting to consider? Thank you!

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notthe rational function field $\mathbf F_q(t)$ but Laurent series fields such as $\mathbf F_q((t))$ and $\mathbf F_q((1/t))$. $\endgroup$ – KConrad Oct 18 '14 at 17:40