Euclid and finite fields In 300 BC or so Euclid pointed out that if $S$ is any finite set of prime numbers then the prime factors of $1+\prod S$ are not in $S$, so that $S$ can always be extended to a larger finite set.
Much later (19th century?  20th?) someone (who?) pointed out that if $F$ is a finite field then $1+\prod\limits_{\alpha\in F}(x-\alpha)$ has no zeros in $F$, so we must extend $F$ to a larger finite field if we want all polynomials to have zeros.
What interesting general propositions are these both special cases of?
 A: If $S$ is any finite set of nonunits of a commutative ring $R$, then $1+\prod S$ is not divisible by any element of $S$. Thus if $R$ is atomic, unless $1+\prod S$ is a unit then the set $S$ fails to contain all irreducibles. (I had originally failed to consider the possibility that $1+\prod S$ could be a unit. As Bill Dubuque notes, we need to amend our reasoning to get around this obstacle. For $\Bbb Z$ and $\Bbb F_q[x]$ one can choose positive naturals and use absolute value or use degrees respectively to verify that $1+\prod S$ is a nonunit.)
Specializing, if $\frak P$ is any set of irreducibles (say, of a certain type, like "degree one polynomials") and $S$ any finite subset of $\frak P$, and $1+\prod S\not\in R^\times$ then either $S$ is proper or not all elements of $R$ are have a divisor in $\frak P$. If $F$ is a finite field then we know set set $\frak P$ of linear polynommials is finite, so this argument shows there are polynomials without linear factors, equivalently without roots. In any ring primes are a certain type of irreducible element, which gives Euclid's conclusion for $\Bbb Z$.
A: 
"Early in the development of the subject it was noticed that $\Bbb{Z}$ has many properties in common with $A=\Bbb{F}[T]$, the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, bothe have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold in $\Bbb{Z}$ have analogues in $A$."

That quote is from the preface of Michael Rosen's book Number Theory in Function Fields, and is a central theme in that book.
The early parts of that book are IMHO very accessible. The analogues of for example the following concepts/results are explained:


*

*Euler totient function

*Wilson's theorem 

*Prime number theorem (the result $\pi(x)\approx \dfrac{\ln x}x$ in $\Bbb{Z}$)

*Basic arithmetic functions


These rely on the idea that the degree of a polynomial is an adequate measure of its size much like the absolute value of an integer.
Reciprocity laws are also easy to derive in the ring $A$. If $Q$ and $P$ are monic irreducibles of respective degree $m, n$ in $A$, then we have the symmetric function
$$
\prod_{\alpha,\beta}(\beta-\alpha)=(-1)^{mn}\prod_{\beta,\alpha}(\alpha-\beta),
$$
where $\alpha$ (resp. $\beta$) ranges over zeros of $Q$ (resp. $P$). The reciprocity laws follow from this.


*

*The theory of $L$-functions is also easier in $A$ (the analytical problems are easier to handle), and leads to the equidistribution of irreducible monic polynomials into (coprime) residue classes modulo a given polynomial - a perfect analogy with Dirichlet's result on equidistribution of prime numbers into cosets modulo a given integer. Several people have found very explicit results in interesting enough subcases.

*The study of extension fields of function fields (an analogue of algebraic number theory) is a natural extension. There is an analogue of the Riemann hypothesis - introduced in Rosen's chapter 4, and proven(!) in chapter 8. Algebraic geometric thinking and tools help here, but Rosen's book is IIRC self-contained to this extent.


Other analogues studied in Rosen's book include:


*

*Artin's primitive root Conjecture (proven by Bilharz)

*ABC-conjecture (a Theorem on the function field side)

*Material leading to class field theory (this is more advanced)


An analogue that I know to have been studied, but was left out from Rosen's book is the game related to twin primes. The analogue here depends on the size of the field $|\Bbb{F}|=q$. Remember that we view low degree polynomials as small, so two polynomials are close to each other, if their difference has low degree.


*

*If $q>2$, then a natural analogue of the twin prime conjecture is to ask about the existence of irreducible polynomials differing only at their constant term.

*If $q=2$, then all irreducible polynomials other than $T$ have constant term equal to $1$. Furthermore, the number of terms of an irreducible has to be odd (otherwise $T-1$ is a factor). Thus the smallest possible difference between two irreducible monic polynomials is $T^2+T$.


I know that at least Stephen D. Cohen (Glasgow/Bristol) has pursued this, but IIRC the main question is still open.

In general number theory in $A$ seems to be easier than in $\Bbb{Z}$. I'm not the right person to explain the reason for that. Viewed in one way (IIRC Milne's lecture notes?) we get that the usual zeta-function is that of a single point, and when dealing with $A$ only need to deal with the extra layer brought about by a single prime (the direct analogue of the zeta-function looks a lot like a single factor in the Euler product version of Riemann zeta), and that is easier.
