Solving systems of equations

Question

I had a system of equations and i want know the perfect method to solve that:

Solve for $X, Y, Z$ where : $\\$

$X^² = Y + a$

$Y^² = Z + a$

$Z^² = X + a$

Answer 1

You can turn it into a single polynomial $$X=Z^2-a\\Y=(Z^2-a)^2-a\\((Z^2-a)^2-a)^2=Z+a$$
This is a degree-8 polynomial in $Z$, so there isn't a formula for its solution, but software can solve it.
Two of the solutions are the solutions to $Z^2=Z+a$, which can be solved with the quadratic formula. The other six solutions form triples $Z,Z^2-a,(Z^2-a)^2-a$

According to Galois theory, the symmetry that the triples have makes it simpler. They obey cubics whose coefficients are quadratic in $a$. If $b$ and $c$ are the roots of $x^2+x+2=a$, then I think the original octic factors in this way:

$$(Z^2-Z-a)(Z^3+(1+b)Z^2+(b-a)Z-1-ab).\\(Z^3+(1+c)Z^2+(c-a)Z-1-ac)$$

So you can get exact solutions as a function of $a$, if you know how to solve cubics.