Here is the Exercise 2.2.3 from An Invitation to Algebraic Geometry.
Let the group $S_{3}$ act on the polynomial $\mathbb C[x,y,z]$. Find the ring of invariant polynomials.
Since $S_{3}$ contains only 6 elements, it is not hard to solve the problem by considering all permutations. However, the exercise is under chapter Hilbert's Basis Theorem. I don't know how they are related.
So, my question is, is there any way to solve this problem quickly by using the theorem?