I have a book that proves the three laws of quadratic reciprocity for primes i.e $\left(\frac{-1}p\right)=1$ if $p=1\mod4$ and $-1$ if $p=3\mod4$, where $p$ is prime, etc.
But in the end the book says that the three laws are also true for arbitary odd integers.
Why is that true?
Thank you.