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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3$\begingroup$ A possible duplicate: mathoverflow.net/questions/1367/… $\endgroup$– KConradCommented Oct 18, 2014 at 17:35
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6$\begingroup$ One way number theory in the function field case ($\mathbf F_q(t)$ and its finite extensions) can be applied back to more classical problems is in strong estimates on classical exponential sums. The best estimates are usually obtained from analogues of the Riemann hypothesis for zeta-functions or $L$-functions in function fields. By the way, in your question, the analogue of $\mathbf R$ in the function field case is not the rational function field $\mathbf F_q(t)$ but Laurent series fields such as $\mathbf F_q((t))$ and $\mathbf F_q((1/t))$. $\endgroup$– KConradCommented Oct 18, 2014 at 17:40
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3$\begingroup$ Peter Scholze's ongoing course at Berkeley is a striking example of this analogy, using insights and constructions from the function field case to develop new ideas and techniques in the characteristic 0 case (and perfectoid spaces have opened up striking ways to directly use char. p geometry to solve problems in char. 0). The use of Ngo's results on the Fundamental Lemma in char. p to solve the analogous problem in char. 0 is another fantastic example (deeply ramified extensions in char. 0 can be "approximated" by analogous extensions in char. p). $\endgroup$– user27920Commented Oct 18, 2014 at 20:15
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4$\begingroup$ The Ax-Kochen theorem is an early example of how one can infer theorems on number fields from the analogous ones on function fields. The wikipedia page has a good summary: en.wikipedia.org/wiki/Ax%E2%80%93Kochen_theorem $\endgroup$– Peter ArndtCommented Oct 18, 2014 at 23:22
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$\begingroup$ Thank you very much for all the comments! They are very interesting and helpful! Thank you very much! $\endgroup$– SJYCommented Oct 19, 2014 at 1:24
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