Is to possible to prove the problem with elementary approach as used to prove the case $4n+3$. Most of the proof that proves Infinitude of primes of the form $4n+1$ uses the some theorem from quadratic reciprocity.
So I was curious to know whether this proof can also be done as the same way as of the proof for the case $4n+1$ without using any special result.
I am aware of the proof of this fact available in this site. But I just want the proof in the way this book mentioned.
Any help would be appreciated. Thanks in advance.
The easiest way would be the one highlighted above. The main tool you use to show the infinitude of primes of the form $4n+3$ is the fact that if a number $x$ satisfies: $$x \equiv 3 \pmod{4}$$ then there exists a prime divisor which is $3 \bmod{4}$. This isn't true when you replace $3 \bmod{4}$ by $1 \bmod{4}$.
Thus, we have to use the first supplement of quadratic reciprocity to prove the infinitude of primes which are $1 \bmod{4}$. This is one of the simplest ways we can prove the same.