3.10

1. If the Jacobi symbol is +1, it does not imply that a is a square mod n, but if it is -1, it implies that it definitely is not. Is that correct?

2. Whoa. The law of quadratic reciprocity. In some cases, this can be extended to even powers that are themselves powers of two, right? Suppose we want to determine whether b is a fourth mod p. Then x^4=b (mod p) is the same as (x^2)^2=b (mod p) so we determine, if p=3 (mod 4) (?), x^2. We then determine whether x^2 is a square mod p. Here when we write x^2, we aren't actually assuming that such an x exists, although it looks like it. If necessary, we could let y=x^2 and solve y^2=b (mod p) and then x^2=y (mod p) so that x^4=b (mod p). Is this right?