Intro to Classical Number Theory I am having trouble understanding page 53, of A Classical Introduction to Modern Number Theory, by Kenneth Ireland and Michael Rosen.  
Corollary 3. $$(-1)^{(p-1)/2} = \left(\frac{-1}{p}\right)$$  Where the right-hand side is the Legendre symbol.  
The part I'm tripping over is the next paragraph, which states: Corollary 3 is interesting.  Every odd integer has the form $4k + 1$ or $4k + 3$.  (I understand that).  Using this, one can restate the corollary as:
$$x^{2}\equiv -1\bmod(p)\text{ has a solution} \iff p \text{ is of the form } 4k + 1$$
How did the authors go from the corollary to the restatement? 

The congruence has a solution iff the Legendre symbol is equal to $1$ which is the case iff the power of $-1$ in the explicit formula for the Legendre symbol is even. 
