I have the following problem, given the definition:
Let $p$ be an odd prime, and let $S$ be a set of $(p-1)/2$ integers. We call $S$ gaussian set modulo $p$ if $S\cup-S=S\cup\{-s\mid s\in S\}$ Is a reduced system of residues modulo $p$.
Reading the proof of a theorem uses that result. But the result is not explained.
Let $p$ be an odd prime. Prove that $\{1,2,...,(p-1)/2\}$ is a Gaussian set modulo $p$.
Note that if $T$ is a reduced system of residue then $T'$ will also be where $T'$ arises from $T$ by adding $p$ to some elements. So $S\cup -S$ is a reduced system iff $S\cup (-S+p)$ is. Now we have $S\cup (-S+p)=\{1,2,\dots,p-1\}$. Since $p$ is prime the elements of this set are all coprime to $p$ and it contains $\phi(p)=p-1$ incongruent elements. Therefore it is a reduced system of residues mod $p$.