Question

Find a square root of −1 modulo p for each of the primes p = 17 and p = 29. Does −1 have a square root modulo 19? Why or why not?

Answer 1

We need to find x such that, x² + 1 0 (mod 17)

Then, x √(-1) (mod 17)

x = 4 (mod 17) satisfies, 4² + 1 17 0 (mod 17)

So, √(-1) (mod 17) = 4 (mod 17)

We need to find, h such that, y² + 1 0 (mod 29)

Then, y √(-1) (mod 29)

y = 12 (mod 29) satisfies, 12² + 1 = 145 0 (mod 29)

So, √(-1) (mod 29) = 12 (mod 29)

-1 doesn't have a square root modulo 19, since, 19 is a prime of the form 4k+3