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.