Find the minimal polynomial with real root

In summary, the conversation discusses finding the minimal polynomial with a root of 21/3 + 21/2. The polynomial is found and verified, but the speaker does not have a method for determining irreducibility. They suggest trying substitutions or computing the dimension of the vector space. The conversation also mentions the rational root test as a potential method.
  • #1
Daveyboy
58
0
Find the minimal polynomial with root 21/3 + 21/2.

I would just use maple but I do not have it installed on this machine.
I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in this case. Do you have another method? I have not tried substituting x=x+1 or x=x-1 or other substitutions.

The polynomial is x6-6x4-4x3+12x2-24x-4
 
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  • #2
Try computing [itex][ \mathbb{Q}(\sqrt[3]{2} + \sqrt{2}) : \mathbb{Q} ][/itex].

Or... the dimension of the vector space spanned by powers of [itex]\sqrt[3]{2} + \sqrt{2}[/itex].

Or... something else that would tell you information about the minimal polynomial.
 
Last edited:
  • #3
Daveyboy said:
Find the minimal polynomial with root 21/3 + 21/2.

I would just use maple but I do not have it installed on this machine.
I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in this case. Do you have another method? I have not tried substituting x=x+1 or x=x-1 or other substitutions.

The polynomial is x6-6x4-4x3+12x2-24x-4

You don't need to show the polynomial is irreducible, do you? You just want to show it has no rational roots. Look at the rational root test. If that has a rational root, the root must divide 4.
 

1. What is the definition of a minimal polynomial with real root?

A minimal polynomial with real root is a polynomial with the smallest degree that has a real number as a solution. This polynomial is unique and irreducible, meaning it cannot be factored into smaller polynomials with real coefficients.

2. How do you find the minimal polynomial with real root?

To find the minimal polynomial with real root, you can use either the rational root theorem or the method of undetermined coefficients. Both methods involve setting the polynomial equal to zero and solving for the unknown variable.

3. Can a polynomial have more than one minimal polynomial with real root?

No, a polynomial can only have one minimal polynomial with real root. This is because the minimal polynomial is unique and cannot be factored into smaller polynomials.

4. What is the significance of finding the minimal polynomial with real root?

Finding the minimal polynomial with real root can help determine the algebraic properties of a number with a real root. It can also be useful in solving equations and understanding the behavior of a function.

5. Is it possible for a polynomial to have no minimal polynomial with real root?

No, every polynomial must have a minimal polynomial with real root. This is because every polynomial has at least one real root, even if it is a complex number. The minimal polynomial is the smallest polynomial that has this real root as a solution.

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