Hello I am wondering if any one can help me I am trying to figure out how these below answers where came to too.
Calculate the quadratic residues in Z∗17.
Solution: This can be done by direct calculation. Since the quadratic residues are symmetric, they only need to be calculated for half of the values.
In this case the quadratic residues are: 1, 4, 9, 16,8, 2, 15, 13.
I would grateful if someone could step me though this step by step. As cant seem to get this.
The quadratic residues are simply found by squaring each of the elements in $\mathbb{Z}_{17}$ (that's just what they are): \begin{align} (1)^2&\equiv 1\mod 17\\ (2)^2&\equiv 4\mod 17\\ (3)^2&\equiv 9\mod 17\\ (4)^2&\equiv 16\mod 17\\ (5)^2&\equiv 8\mod 17\\ (6)^2&\equiv 2\mod 17\\ (7)^2&\equiv 15\mod 17\\ (8)^2&\equiv 13\mod 17\\ \end{align} Since $a^2=(-a)^2$ we only have to check until $8$, because $9\equiv -8\mod 17$, $10\equiv -7\mod 17$, etc.. Thus, the quadratic residues are $\{1,4,9,16,8,2,15,13\}$.
Hope this helped!