MHB Find the minimal polynomial of some value a over Q

[E01]

I'm trying find the minimal polynomial of $$a=3^{1/3}+9^{1/3}$$ over the rational numbers. I am currently going about this by trying to construct a polynomial from a (using what I intuitively feel would be a sufficiently small number of operations).
Then I'd show it's irreducible by decomposing it into linear functions with complex functions and show the combination of any of these linear components is not a polynomial with rational coefficients (this part I'm worried about as if the degree is greater than 6 I'm not sure how to find the roots of the equation). Can anyone give me a hint(like what the degree of the minimal polynomial is)?

Okay, I just found a polynomial of degree 27.

 

[Opalg]

I'm trying find the minimal polynomial of $$a=3^{1/3}+9^{1/3}$$ over the rational numbers. I am currently going about this by trying to construct a polynomial from a (using what I intuitively feel would be a sufficiently small number of operations).
Then I'd show it's irreducible by decomposing it into linear functions with complex functions and show the combination of any of these linear components is not a polynomial with rational coefficients (this part I'm worried about as if the degree is greater than 6 I'm not sure how to find the roots of the equation). Can anyone give me a hint(like what the degree of the minimal polynomial is)?

Okay, I just found a polynomial of degree 27.

$a^3 = \bigl(3^{1/3} + 3^{2/3}\bigr)^3 = 3 + 3\cdot3^{1/3}\cdot3^{2/3}\bigl(3^{1/3} + 3^{2/3}\bigr) + 9$ (binomial expansion). That should simplify to a cubic equation for $a$.