Math Help - infinite primes?

1. infinite primes?

hello guys . question here
how can i prove that there exists infinitely many primes p such that p = 3 mod 4.

i have a little inkling as i know that if a,b=1 mod 4 then ab = 1 mod 4. Im guessing it would be along the lines of euclids theorem?

2. Suppose there are only finitely many such primes, p_1, p_2, ..., p_n (where p_1=3). Look at the product $4p_2p_3\cdots p_n + 3$ and think about what its prime factors must look like.

3. The case 4k+1 is more difficult.