High School Mathematics Extensions/Further Modular Arithmetic/Project/Finding the Square Root

*Finding the square root* Legendre Symbol .. to be expanded There is actually a simple way to determine whether a is square Let g be a generator of G where G is the multiplicative group mod p. Since all the squares form a group therefore, if a is a square, then and if a is not a square then we shall use these facts in the next section. .. to be expanded *Finding the square root* We aim to describe a way to find a square root in mod m. Let's start with the simplest case, where p is prime. In fact, for square root finding, the simplest case also happens to be the hardest. If p ≡ 3 (mod 4) then it is easy to find a square root. Just note that if a has a square root then So let us consider primes equivalent to 1 mod 4. Suppose we can find the square root of a mod p, and let With the above information, we can find the square of a mod p2. We let we want x2 ≡ a (mod p2), so for some k as so , we see that so if we need to find such that , we simply need to make the subject . ..generalisation ..example ..method for finding a sqr root mod p