Prove that there are infinitely many primes of the form $5k+3$

I known this is Dirichlet's Theorem Special case,but I want to find a similar following elementary proof:

1:there are infinitely many primes of the form $8k+3$,consider $x^2+2\equiv 0\pmod p$

2: there are infinitely many primes of the form $5k+4$,consider $x^2-5\equiv 0\pmod p$

3: there are infinitely many primes of the form $7k+6$,consider $x^3+x^2-2x-1\equiv 0\pmod p$

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