I found that the Legendre symbol or quadratic character is a function introduced by Adrien-Marie Legendre in 1798. He created this during his partly successful attempt to prove the law of quadratic reciprocity. We also always talk about whether something is just multiplicative of completely multiplicative. So, I thought it was interesting that it is a completely multiplicative function in its top argument. This property can be simply explained in the fact that the product of two residues or non-residues is a residue, whereas the product of a residue with a non-residue is a non-residue.

I also found some similar rules we talked about in class, except when we have a = -3 or 3 = 5:

(-3/p)= 1 if p is congruent to 1 mod 6

-1 if p is congruent to 5 mod 6

(5/p)=1 if p is congruent to 1,9 mod 10

-1 if p is congruent to 3,7 mod 10