MHB Find the minimal polynomial of some value a over Q

Click For Summary
The discussion centers on finding the minimal polynomial of the expression a = 3^(1/3) + 9^(1/3) over the rational numbers. The user initially attempts to construct a polynomial and demonstrate its irreducibility by analyzing its linear components with complex functions. They express concern about the degree of the polynomial, which they initially found to be 27, and the difficulty in finding roots for higher degrees. However, they later derive a cubic equation for a using binomial expansion, suggesting a more manageable approach to determining the minimal polynomial. The focus remains on establishing the correct polynomial and proving its irreducibility.
E01
Messages
8
Reaction score
0
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.
 
Last edited:
Physics news on Phys.org
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 $a$.
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
View full post »

Similar threads

Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
2K
MHB The polynomial is irreducible over Q(i)
  • · Replies 16 ·
Replies
16
Views
4K
Undergrad Looking for a creative or quick method for finding roots in GF(p^n)
  • · Replies 3 ·
Replies
3
Views
2K
MHB The irreducible polynomial is not separable
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What can be deduced about the roots of this polynomial?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Find Min Polynomial of $\alpha$ Over $\mathbb{Q} | Solution Included
  • · Replies 1 ·
Replies
1
Views
3K
MHB How can we prove that E/Q is a normal extension?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Splitting ring of polynomials - why is this result unfindable?
  • · Replies 9 ·
Replies
9
Views
2K
MHB Eisenstein polynomial and field extension
  • · Replies 24 ·
Replies
24
Views
5K
Finding the minimal polynomial over Q
  • · Replies 27 ·
Replies
27
Views
6K