Algorithms (textbook) by Dasgupta. Prob. 1.36 Square Roots

Question

To begin I want to say that this is NOT a homework problem. I know stackoverflow condemns people who ask for homework solutions. I merely doing this problem out of interest.

This is the question that I am working on:

Need help with part (b), not (a)

I believe I understand (a); I had my own answer but I managed to compare my solution with a Chegg preview solution (it doesn't show part (b)). So far from my understanding of part (b) is the following:

when they say

x is a square root of a modulo p if a = x^2(mod p)

they mean: x = sqrt(a mod p) IF a = x^2(mod p).

Now, where it says,

if a has a square root modulo p, then a^((p+1)/4) is such a square root

confuses me a lot. I'm not really sure what this line means!

Answer 1

if a has a square root modulo p, then a^((p+1)/4) is such a square root

=

If there exists K such that K^2 mod p = a ,

then

a^((p+1)/4) mod p = K