Cubic Reciprocity, solving $x^3 \equiv 19 \pmod{43}$ I know that there is no solution for the congruence $x^3 \equiv 19 \pmod{43}$, but I don't know how to go about showing it. I know I should use cubic reciprocity, but I'm not sure how that works. Can somebody walk me through how to show this?

Cubic reciprocity says your equation is solvable iff $x^3 \equiv 43 \mod 19$, or $x^3\equiv 5 \mod 19$ is solvable.
One of Euler's conjectures states that $5$ is a cubic residue of $p$ iff $15|b$, or $3|b$ and $5|a$, or $15 | (a \pm b)$, or $15|(2a \pm b)$. Here $19 = a^2 + 3b^2$, and this conjecture applies because $19$ is a prime $1 \mod 3$.  
Solving, $a = 4, b=1$. None of the conditions hold, so the equation has no solutions. 
