# The map $\left(\frac{.}{p}\right):=\mathbb F_p \rightarrow [\pm1]$ taking $a\in \mathbb F_p$ to $\left(\frac{a}{p}\right)$ is homomorphism

Prove that the map $\left(\frac{.}{p}\right):=\mathbb F_p \rightarrow [\pm1]$ taking $a\in \mathbb F_p$ to $\left(\frac{a}{p}\right)$ is a homomorphism

I don't know how to even start this problem. I would greatly appreciate it if someone could help me.

• There are three things to show: the product of two quadratic residues is a quadratic residue, the product of two non-residues is a residue, and the product of a residue and a non-residue is a non-residue. – Slade Dec 17 '15 at 15:52
• @Slade Ok thank you, that really helps – GRS Dec 17 '15 at 15:57
• The Legendre symbol is simply the Character of the group corresponding to -1 i.e. a map from the field to a subset of roots of 1. – Yash V. Singh Dec 17 '15 at 17:18