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Prove that if p is a prime number of the form 4n + 1

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Prove that if p is a prime number of the form 4n + 1,
then the following congruence holds:

x^2 is congruent to -1 mod p.

The more detail problem is in the attached file.

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The solution is in the attached file.

Ring Theory 3
Solving Congruences
By :- Thokchom Sarojkumar Sinha

Prove that if p is a prime number of the form 4n + 1
then the following congruence holds,

Proof :- Suppose that

Since p = 4n + 1 then p - 1 = 4n
which implies that

Therefore the product for ...

Solution Summary

The solution explains the congruence: x^2 is congruent to -1 mod p.
The solution is given in detail.
This is mainly for solving the congruence by using the Wilson's theorem.

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