# What is the quadratic character of the field over which elliptic curve is defined?

I'm trying to understand the injective encoding of a message to an elliptic curve point (from this paper).

However, I'm not sure what do they mean by the quadratic character of the field. Do you know what does it mean and how to compute it? Is it somewhat similar to the Jacobi symbol?

For any finite field $$\mathbb{F}_q$$ of odd characteristic, there is a unique nontrivial morphism $$\mathbb{F}_q^*\to\{-1,1\}$$: it takes the value $$1$$ on squares and $$-1$$ on nonsquares (and it is usually extended to $$0$$ on $$0$$). This is the Legendre symbol when $$q$$ is prime, but it's called the quadratic character in general.