Legendre symbol simplification I saw a simplification using the Legendre symbol which said $$\left(\dfrac{19}{29}\right) = \left(\dfrac{10}{19}\right) = \left(\dfrac{2}{19}\right) \cdot \left(\dfrac{5}{19}\right) = -1.$$ My question is how did they get $\left(\dfrac{19}{29}\right) = \left(\dfrac{10}{19}\right)$ and finally that $\left(\dfrac{2}{19}\right) \cdot \left(\dfrac{5}{19}\right) = -1$?

For the first question
$$
\left(\dfrac{19}{29}\right) = \left(\dfrac{29}{19}\right)
 = \left(\dfrac{10}{19}\right)
$$
since $29$ is $1$ mod $4$.
For the second, 
$$
\left(\dfrac{5}{19}\right) = 
\left(\dfrac{19}{5}\right) = 
\left(\dfrac{-1}{5}\right) = 1
$$
since $5$ is $1$ mod $4$
and
$$
\left(\dfrac{2}{19}\right) = -1
$$
because $19$ is $3$ mod $8$. (http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/the-quadratic-character-of-2)
