TITLE: Dirichlet's theorem on primes in arithmetic progression
QUESTION [11 upvotes]: Is there a proof in the spirit of Euclid to prove Dirichlet's theorem on primes in arithmetic progression? (By the spirit of Euclid, I mean assuming finite number of primes we try to construct another number which has a prime factor which falls in the same equivalence class as the other primes but the number is not divisible by any of the primes we considered in the initial list.)
I am aware of the proof using $L$ functions but I am curious to know if Euclid's "simple" idea can be extended to all other cases as well. I tried googling but am unable to find a proof other than the ones relying on $L$ functions. If it is not possible, to what cases can Euclid's idea be extended to?
Any other proofs are welcome as well.

REPLY [12 votes]: As Jonas says, Keith Conrad's paper will tell you that the answer is essentially no.  He defines a certain notion of "Euclidean" proof coming from writing down a polynomial with certain properties and gives two classic results which say that these proofs exist for primes in arithmetic progression $a \bmod n$ if and only if $a^2 \equiv 1 \bmod n$.
The first case these proofs can't handle is primes congruent to $2 \bmod 5$ (equivalently, primes congruent to $3 \bmod 5$).  The basic problem is that there is no way to force a positive integer to have factors congruent to $2 \bmod 5$ and not congruent to $3 \bmod 5$ (or the other way around) solely by controlling its residue class $\bmod 5$.  In more sophisticated language, $2$ and $3$ lie in all the same subgroups of $(\mathbb{Z}/5\mathbb{Z})^{\ast}$.

REPLY [6 votes]: A couple more references:
Ram Murty and Nithum Thain,
Prime numbers in certain arithmetic progressions,
Paul Pollack, 
Hypothesis H and an impossiblity theorem of Ram Murty.