How can I compute the Galois group of $x^4+7x+7$ over $\mathbb{Q}$? I believe it can be done using some general results about discriminants and cubic resolvants such as in this document: https://kconrad.math.uconn.edu/blurbs/galoistheory/cubicquartic.pdf#page6
However I would prefer to do a more “direct” computation. By direct I mean something not using discriminants (or the general formulas for a quartics roots either). Usually though this involves being able to deduce what some of the roots are explicitly and I don’t know how to do this for this particular polynomial. Maybe this approach isn’t viable, if that’s the case then at least knowing that would be helpful. Thanks in advance.