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Cross-posted from Math.SE.

I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.

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  • $\begingroup$ "almost all" polynomials are irreducible, so your observations on $(x^2+2x+1)p(x)+x$ should be no surprise. $\endgroup$
    – Gerry Myerson
    Commented Nov 21, 2019 at 21:38
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    $\begingroup$ It’s funny you ask this. Yesterday my grad student told me of a result that might fit what you’re looking for: Ehrenfeucht’s criterion says if $K$ is a field of characteristic zero, $n\ge3$ and $f_1,/ldots,f_n$ are polynomials of degree at least 1 then the polynomial $f_1(x_1)+\ldots f_n(x_n)$ is irreducible in $K[x_1,\ldots,x_n]$. $\endgroup$
    – Anthony Quas
    Commented Nov 22, 2019 at 21:23

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This is intended as a long comment.

There is of course a well-known similarity between the polynomial ring $\mathbb F_p[x]$ and the ring of integers $\mathbb Z$, and more generally between a curve over finite field and the $\operatorname{Spec}$ of the ring of integers of a number field.

Some of the number theory problems do translate to the polynomial ring case (and is often simpler there). For example, it is easy to show the "prime number theorem", i.e. the number of irreducible polynomials of degree $d$ is about $\frac{1}{d}$ of all polynomials of degree $d$.

There is also a parallel "Goldbach's conjecture" for $\mathbb F_2[x]$: any polynomial in $x\mathbb F_2[x]$ with degree $> 1$ is the sum of two irreducible polynomials.

Note however that these are examples in additive number theory, which a priori are different from problems arising in the geometric Langlands (for example, quadratic reciprocity law would fall in this latter class).

Nevertheless, your claim that $(x^2 + 2x + 1)p(x) + x$ is irreducible for "almost all" $p(x)$ doesn't make sense to me, in view of the "prime number theorem".

In fact, among all polynomials $p(x) \in \mathbb F_2[x]$ with non-zero constant term and of degree $\leq 15$, there are only $8273$ out of a total of $32768$ that are irreducible.

If you are talking about polynomials over $\mathbb Z$ instead of $\mathbb F_p$, then I'd say there are not much comparable: we go from dimension $1$ to dimension $2$.

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  • $\begingroup$ Thank you for the comprehensive answer--I was unaware of the Goldbach analog in $\mathbb{F}_{2}[x]$ and perhaps it should've been a first search :). And whoops, I wasn't very careful with what I meant by my example. Firstly, I'm working in $\mathbb{Z}[x]$ and I just plugged in a few polynomials manually, so not really the best test. But I guess generally, I'm interested in what we can concretely say about polynomials of that form as far as irreducibility goes (conditions on $p(x)$ that guarantee irreducibility, density of irreducible polynomials in $\mathbb{Z}[x]$, etc). $\endgroup$
    – Dan
    Commented Nov 21, 2019 at 21:03

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