Find the minimal polynomial of some value a over Q

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SUMMARY

The minimal polynomial of the value \( a = 3^{1/3} + 9^{1/3} \) over the rational numbers \( \mathbb{Q} \) is determined to be of degree 27. The user is constructing this polynomial through operations involving \( a \) and aims to demonstrate its irreducibility by decomposing it into linear functions with complex coefficients. The discussion highlights the use of binomial expansion to simplify \( a^3 \) into a cubic equation, which is a critical step in finding the minimal polynomial.

PREREQUISITES
  • Understanding of minimal polynomials in field theory
  • Familiarity with binomial expansion techniques
  • Knowledge of polynomial irreducibility criteria
  • Basic concepts of algebraic numbers and their properties
NEXT STEPS
  • Study the properties of minimal polynomials over \( \mathbb{Q} \)
  • Learn about polynomial irreducibility tests, such as Eisenstein's criterion
  • Explore the use of Galois theory in determining the degree of extensions
  • Investigate cubic equations and their roots in the context of algebraic numbers
USEFUL FOR

Mathematicians, algebra students, and anyone interested in field theory and polynomial equations, particularly those studying algebraic numbers and their minimal polynomials.

E01
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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.
 
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E01 said:
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 $a$.
 

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