Determine whether the following integers are quadratic residues of the prime 131:
(a) 14
(b) 12
(c) 15
(d) 65
(a) We want to calculate the Legendre symbol
$$ \begin{align} (\frac{14}{131}) & = (\frac{2}{131}) \times (\frac{7}{131}) \\ \\ & = (-1) \times (\frac{7}{131}) \tag{Proposition 7.15} \\ \\ & = -1 \times (\frac{131}{7}) = -1 \times -(\frac{5}{7}) \tag{Corollary 7.17} \\ \\& = (\frac{7}{5}) = (\frac{2}{5}) \tag{Corollary 7.17} \\ \\ & = -1 \tag{Proposition 7.15} \end{align}$$
And so 14 is not a quadratic residue of prime 131.
(b) We want to calculate the Legendre symbol
$$ \begin{align} (\frac{12}{131}) & = (\frac{2^2}{131}) \times (\frac{3}{131}) = 1 \times (\frac{3}{131}) \tag{$2^2$ is a quadratic residue} \\ \\ & = - (\frac{131}{3}) = -(\frac{2}{3}) \tag{Corollary 7.17} \\ \\ & = -(-1) = 1 \tag{Proposition 7.15}\end{align}$$
And so 12 is a quadratic residue of prime 131.
(c) We want to calculate the Legendre symbol
$$ \begin{align} (\frac{15}{131}) & = (\frac{3}{131}) \times (\frac{5}{131}) = (1) \times (\frac{5}{131}) \tag{result from (b)} \\ \\ & = (\frac{131}{5}) = (\frac{1}{5}) = 1 \tag{Corollary 7.17} \end{align}$$
$$ \begin{align} (\frac{65}{131}) & = (\frac{5}{131}) \times (\frac{13}{131}) = (1) \times (\frac{13}{131}) \tag{result from (c)} \\ \\ & = (\frac{131}{13}) = (\frac{1}{13}) = 1 \tag{Corollary 7.17} \end{align}$$
And so 65 is a quadratic residue prime 131.