How can I approach a question of the form
Can there exist an element β in Zx satisfying $$ β^a= b $$
and $$ β^c= d $$?
I understand how to find primitive elements of Zx and could use it to find β if I was given just 1 of these equations that have to be satisfied, but don't see how that helps me as β doesn't necessarily have to be primitive to satisfy them.
Is there any way I can approach this without brute forcing this?
Thanks
Note: I know everything except β not looking for an answer just the approach
A general approach that is quite powerful here is to consider the gcd of $a$,$c$ — or more generally, that of $a$,$c$,$\phi(x)$ (assuming that $a$,$c$ don’t already divide $\phi(x)$, which is hard to tell from the information you’ve suppressed).
Suppose $g$ is this gcd. Then there exists integers $r,s$ such that
$$ra + sc = g.$$
This allows you to combine the two equations into their lowest-degree form:
$$\beta^g = b^r d^s.$$
First, you can do a consistency check: if the RHS evaluates to $m$, does $m^{a/g} = b$ and $m^{c/g} = d$? If not, then the equations are clearly inconsistent.
If it does pass these tests then you have reduced the two equations down to one that is easier to check for (it has lower degree).
Assuming that $b,d$ are relatively prime to $x$ (again hard to say due to the suppression of details), then you can also combine this with the third equation $\beta^{\phi(x)} = 1$.