Evaluate the following Legendre symbols $(\frac{a}{p})$ where $p$ is prime in each case:
(a) $(\frac{12}{ 71} )$
(b) $(\frac{15}{ 101} )$
(c) $(\frac{28 }{163} )$
(d) $(\frac{75 }{ 541} )$
(e) $(\frac{360 }{1223} )$
(f) $(\frac{115 }{1987} )$
(g) $(\frac{700 }{3571} )$
(h) $(\frac{703}{ 4409} )$
[Hint: 703= 19 × 37.]
These exercises are practice in using the new proposition and lemmas introduced in section 7.3.
(a) We proceed as follows
$$ \begin{align} (\frac{12}{71}) & = (\frac{2^2 \times 3}{71}) \\ \\ & = (\frac{3}{71}) \tag{$2^2$ is a quadratic residue} \\ \\ & = -(\frac{71}{3}) \tag{Corollary 7.17}\\ \\ & = -(\frac{2}{3}) \tag{$71\equiv 2 \pmod 3$} \\ \\ & = -(-1) = 1 \tag{Proposition 7.15} \end{align} $$
(b) We proceed as follows
$$ \begin{align} (\frac{15}{101}) & = (\frac{3 \times 5}{101}) \\ \\ & = (\frac{3}{101}) \times (\frac{5}{101}) \\ \\ & = (\frac{101}{3}) \times (\frac{101}{5}) \tag{Corollary 7.17} \\ \\ & = (\frac{2}{3}) \times (\frac{1}{5}) \\ \\ & = -1 \times 1 = -1 \tag{Proposition 7.15 and 7.11} \end{align} $$
(c) We proceed as follows
$$ \begin{align} (\frac{28}{163}) & = (\frac{2^2 \times 7}{163}) \\ \\ &= (\frac{7}{163}) \tag{$2^2$ is a quadratic residue} \\ \\ & = - (\frac{163}{7}) \tag{Corollary 7.17} \\ \\ & = -(\frac{2}{7}) \\ \\ & = -1 \tag{Corollary 7.15} \end{align} $$
(d) We proceed as follows
$$ \begin{align} (\frac{75}{541}) & = (\frac{5^2 \times 3}{541}) \\ \\ & = (\frac{3}{541}) \tag{$5^2$ is a quadratic residue} \\ \\ & = (\frac{541}{3}) \tag{Corollary 7.17} \\ \\ & = (\frac{1}{3}) = 1 \end{align} $$
(e) We proceed as follows
$$ \begin{align} (\frac{360}{1223}) & = (\frac{2^2 \times 2 \times 3^2 \times 5}{1223}) \\ \\ & = (\frac{2}{1223}) \times (\frac{5}{1223}) \tag{$2^2$ and $3^2$ are quadratic residues} \\ \\ & = (1) \times (\frac{5}{1223}) \tag{Corollary 7.15} \\ \\ & = (\frac{1223}{5}) \tag{Corollary 7.17} \\ \\ & = (\frac{3}{5}) = (\frac{5}{3}) \tag{Corollary 7.17} \\ \\ & = (\frac{2}{3}) = -1 \tag{Proposition 7.15} \end{align} $$
(f) We proceed as follows
$$ \begin{align} (\frac{115}{1987}) & = (\frac{5}{1987}) \times (\frac{23}{1987}) \\ \\ & = (\frac{1987}{5}) \times -(\frac{1987}{23}) \tag{Corollary 7.17} \\ \\ & = (\frac{2}{5}) \times -(\frac{3^2}{23}) \\ \\ & = (-1) \times -(1) = 1 \tag{Proposition 7.15 and $3^2$ is QR} \end{align} $$
(g) We proceed as follows
$$ \begin{align} (\frac{700}{3571}) & = (\frac{2^2 \times 5^2 \times 7}{3571}) = (\frac{7}{3571}) \tag{$2^2$ and $5^2$ are QR} \\ \\ & = - (\frac{3571}{7}) = -(\frac{1}{7}) = -1 \tag{Corollary 7.17} \end{align} $$
(h) We proceed as follows
$$ \begin{align} (\frac{703}{4409}) & = (\frac{19}{4409}) \times (\frac{37}{4409}) \\ \\ &= (\frac{4409}{19}) \times (\frac{4409}{37}) = (\frac{1}{19}) \times (\frac{6}{37}) = (1) \times (\frac{6}{37}) \tag{Corollary 7.17} \\ \\ & = (\frac{2}{37}) \times (\frac{3}{37}) = (-1) \times (\frac{3}{37}) \tag{Proposition 7.15} \\ \\ & = -(\frac{37}{3}) = -(\frac{1}{3}) = -1 \tag{Corollary 7.17} \end{align} $$