Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to $\mathbb{F}_q^*$ lifted to $\mathbb{Z}$ in the usual way. Could you recommend any good book on this?

I wonder about the finite-field analogies to classical facts with complex numbers. Beyond the obvious fundamentals, it would be interesting to know, say, what are the primitive characters here, the number of these, conditions on a character so that its associate Gauss sum is separable, Fourier analysis, Heisenberg uncertainty, etc.

Regarding separability, I've noticed for instance that the Gauss sum attached to the principal Dirichlet character (essentially Ramanujan's sum) seems to be separable in the case when the characteristic of the field is $2$ and the modulus of the character is square-free; so this principal character over this field "behaves" like a primitive character. Things of the sort... Thanks a lot.

Definition of separability: For $b \in \mathbb{Z}$ and a Dirichlet character $\chi$ modulo $N$, its associate Gauss sum is given by $$G(b, \chi) = \sum_{x \in \mathbb{Z}_N} \chi(x) e^{2\pi i xb/N}.$$ It is called separable if $$G(b, \chi) = G(1, \chi) \overline{\chi(b)}$$ for all $b \in \mathbb{Z}$. Note that $\chi$ is primitive (not induced by a character of modulus a strictly smaller divisor of $N$) if and only if its associate Gauss sum is separable.

• The group $\Bbb{F}_q^*$ is cyclic of order $q-1$. Therefore homomorphisms from any group to $\Bbb{F}_q^*$ are (more or less) the same thing as homomorphisms to $\langle e^{2\pi i/(q-1)}\rangle$. I don't know what you mean by a separable Gauss sum. – Jyrki Lahtonen Jul 4 '15 at 4:39
• Hi Jyrki. Although I agree with you that most of the theory will be more or less the same, my gut feeling is that technical differences will make an impact here and there. For instance, this example with Ramnujan sum when $\text{char}(\mathbb{F}_q) = 2$. Also for example the proof of Heisenberg's uncertainty in Fourier analysis particularly uses the norm on complex numbers. So what would be the correct norm here on the finite field, etc? – user152169 Jul 4 '15 at 19:31