Find the minimal polynomial of some value a over Q

In summary, the conversation involves finding the minimal polynomial of $a=3^{1/3}+9^{1/3}$ over the rational numbers. The speaker is trying to construct a polynomial with a small number of operations and show its irreducibility. They are worried about finding the roots if the degree is greater than 6. However, they have found a polynomial of degree 27 by using binomial expansion.
  • #1
E01
8
0
I'm trying find the minimal polynomial of \(\displaystyle a=3^{1/3}+9^{1/3}\) 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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  • #2
E01 said:
I'm trying find the minimal polynomial of \(\displaystyle a=3^{1/3}+9^{1/3}\) 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 $a$.
 

FAQ: Find the minimal polynomial of some value a over Q

What is the definition of minimal polynomial?

The minimal polynomial of a value a over Q is the monic polynomial of lowest degree with integer coefficients that has a as a root.

How do I find the minimal polynomial of a value a over Q?

To find the minimal polynomial of a value a over Q, you can follow these steps:
1. Begin with a polynomial of the form x^n + cn-1x^(n-1) + ... + c1x + c0, where n is the degree of the polynomial and c0, ..., cn-1 are integer coefficients.
2. Substitute a for x in the polynomial and solve for c0, ..., cn-1.
3. If a is a root, then the polynomial obtained in step 2 is the minimal polynomial. If not, increase the degree of the polynomial and repeat steps 1 and 2 until a is a root.

Why is finding the minimal polynomial important?

Finding the minimal polynomial of a value a over Q is important because it allows us to express a as a root of a polynomial with integer coefficients. This can be useful in various mathematical calculations and proofs.

Can there be multiple minimal polynomials for a single value a over Q?

No, there can only be one minimal polynomial for a single value a over Q. This is because the minimal polynomial is unique and has the lowest degree possible for a given value a.

Is the minimal polynomial always a monic polynomial?

Yes, the minimal polynomial is always a monic polynomial. This means that the coefficient of the highest degree term is always 1. This is important because it simplifies the polynomial and makes it easier to work with in calculations.

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