There are several posts on finding minimal polynomials: $ \sqrt 2+\sqrt 3$, $3-2^{1/3}-2^{2/3}$, etc.
Here is the problem: finding the minimal polynomial is always easy, but proving it is minimal seems tricky. The $3-2^{1/3}-2^{2/3}$ question relies on the special property of this number: it cannot have a quadratic minimal polynomial.
The $\sqrt 2+\sqrt 3$ post above asks for a proof of the minimality. Two or three of the answers there seem to suggest that we just need to prove that the polynomial $x^4-10x^2+1$ is irreducible over $\mathbb Q$. This relies on brutal force calculations of expanding each possible combination of the linear factors.
I also know linear algebraic methods. We can write done the matrix corresponding to the number and find the minimal polynomial of that matrix, and this time, it is guaranteed to be minimal due to theorems in linear algebra. However, I don't think it is necessary to use linear algebra all the time. In the case of $\sqrt 2 + \sqrt 3$, we can just do simple algebraic manipulations to get the minimal polynomial.
Question: without using linear algebra, do we have any hope of proving the minimality of polynomials like $x^4-10x^2+1$ easily, without doing a lot of calculations? I am looking for a method that applied to all possible number we can give (with square roots and cube roots), not just $\sqrt 2 + \sqrt 3$.