References about identities of Gauss sum

Question

I am reading the paper. In the end of page 10, there are the following identities of Gauss sum. \begin{align} & h(b) h(a+b) = q^b h(b) h(a), \\ & h(b) g(a+b) = q^b h(b) g(a), \\ & g(a+b) h(a) h(b) = h(a+b) g(a) g(b) + h(a+b) g(a+b), \\ & h(a)^2 = g(a) h(a) + q^a h(a). \end{align}

Are there some references about the proof of these identities? Thank you very much.

Edit: Here the Gauss sum is defined by \begin{align} g(m,c)=\sum_{a \mod c} \left( \frac{a}{c} \right)_n \psi(\frac{am}{c}), \end{align} where $m,c \in \mathfrak{o}_S$, $c \neq 0$, $\left( \frac{a}{c} \right)_n$ is the n-th power residue symbol, and \begin{align} \mathfrak{o}_S = \{x \in F: |x|_v \leq 1 \text{ for } v \not\in S\}, \end{align} $S$ is a fnite set of places of $F$ containing all the archimedean places, $F$ is an algebraic number field containing all the group of $2n$-th roots of unity, $F_S = \prod_{v \in S} F_v$ and $\psi$ is a character of $F$.

$$ h(a) = g(p^a, p^a), \\ g(a)=g(p^{a-1}, p^a). $$

Answer 1

A lot of the papers in that series reference Brubaker and Bump's "Kubota Series paper" for basic facts on these Gauss sums.

If I am not mistaken, in that paper, they reference Neukirch's Algebraic number theory for the relevant questions involving the hilbert symbol and n-power residue symbol.

For me, I have found Davenport's Multiplicative Number theory useful for understanding Gauss sums.

I can't say that all of the identities you list are contained in these references, but the tools to show some them are. I am not sure about the latter two though. That paper is contains several non-standard facts about the Gauss sums, so perhaps it is the only reference to certain identities.