I was watching a video where a problem in Galois theory was posed such that it became necessary to tell if a certain element was a perfect square in a finite field extension of the rationals. By writing a general element in terms of its $\mathbb{Q}$-basis and squaring it, the result was a system of quadratic equations. In particular, it was system of three non-homogeneous equations in three unknowns.
The video deferred the solution to a computer algebra system, which felt unsatisfying. I was wondering what techniques would be useful for solving general equations of this kind.
My only thought was that Groebner bases might be useful (and I'm sure that's how the computer algebra system solves it). But I don't know much about them, and I wonder how amenable the Buchberger algorithm is to solving it by hand.