I am reading [the paper](http://sporadic.stanford.edu/bump/match/gelbart.pdf). 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 $F$.


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



References about identities of Gauss sum
A lot of the papers in that series reference Brubaker and Bump's ["Kubota Series paper"](http://sporadic.stanford.edu/bump/kubota.pdf) for basic facts on these Gauss sums.


If I am not mistaken, in that paper, they reference [Neukirch's Algebraic number theory](http://rads.stackoverflow.com/amzn/click/3540653996) for the relevant questions involving the hilbert symbol and n-power residue symbol.


For me, I have found Davenport's [Multiplicative Number theory](http://rads.stackoverflow.com/amzn/click/0387950974) 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.


