Is there a good textbook that will help me understand the motivation for defining the Legendre symbol (and it's Jacobi generalization), and applications of them to number theory? I have a math degree and I am comfortable with graduate-level texts.
Thank you.
One of the best books I know for these topics is A Classical Introduction to Modern Number Theory by Ireland and Rosen. In particular, see Chapter $5$ on quadratic reciprocity, and for its generalisations in Chapter 9, on cubic and biquadratic reciprocity.