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I suspect this problem is very famous and it must be studied very well. But I searched in Google and I did not find good reference. I will appreciate any answer and reference for any contribution about this question.

Firstly the motivation: We know that if $\gcd(a,b)=1$, then the sequence $an+b$ generates infinite prime numbers, when $n$ varies in the set $\mathbb{N}$. I want to generalize it for irreducible polynomials over $\Bbb{F}_q$, when $q$ is a prime power, i.e, $q=p^\alpha$, where $p$ is a prime number.

Let $g_i(x)\in \Bbb{F}_q[x],$$0\leq i\leq n,$ be irreducible polynomials over the field $\Bbb{F}_q$ and also let $$P(x,y)=\sum_{i=0}^{n}{g_i(x)y^i}.$$

$1)$ Is it true that if $y$ varies in the $\Bbb{F}_q[x]$, then $P(x,y)$ generates infinite irreducible polynomials over $F_q$?

$2)$ for a fixed $q$, is it possible that we choose suitable $g_i(x),$ $g_i(x)\neq g_j(x)$ when $i\neq j,$ such that the first question be true?

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up vote 10 down vote accepted

If $g_1(x)$ and $g_0(x)$ are relatively prime, then there are infinitely many irreducibles of the form $g_1(x) y + g_0(x)$. This is due to Kornblum and Landau (1911) and recorded in many places; see Chapter 4 of Number Theory in Function Fields for a clear modern exposition.

For $G(x,y) = \sum_{i=0}^n g_i(x) y^i$ for $n>1$, the obvious necessary conditions are that $G(x,y)$ be irreducible as a polynomial in $2$ variables, and that there is no $\pi(x)$ such that $\pi(x) | G(x, f(x))$ for all $f(x) \in k[x]$. In the integer case, these conditions are conjectured to be sufficient. In the function field case, they are not! See Conrad and Conrad and Gross. Their exposition is very clear, and they conjecture what sufficient conditions should be.

In a positive direction, Pollack shows that, if you fix $G(x,y)$ in $\mathbb{F}_q[x,y]$ avoiding the obvious obstructions and with $GCD(\deg_y G, q)=1$, fix a degree $n$, and let $y$ range over degree $n$ polynomials with coefficients in $\mathbb{F}_{q^a}$, then the number of prime values obeys the expected asymptotics (and, in particular, is positive) as $a \to \infty$. I am simplifying his result some for exposition; see the original paper.

I don't understand the second part of your question.

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You meant that there are unexpected issues in the function field case, not the number field case. – KConrad Jul 14 '14 at 23:55
Whoops! Thank you. – David Speyer Jul 15 '14 at 1:54
@David, the second question let us to choose carefully the $g_i(x)$, such that $P(x,y)$ generates infinite irreducible polynomial. It is a special case such that we know the conjecture is true for it. Anyway, thanks for your answer. – Shahrooz Janbaz Jul 15 '14 at 6:44

The function field version of Dirichlet's theorem is indeed classical, and much more is known for function fields. You can find the classical theory in Number Theory in Function Fields (Springer Graduate Texts in Mathematics) by Michael Rosen. Chapter 4 is entitled "Dirichlet $L$-Series and Primes in an Arithmetic Progression".

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