Extension Fields: Is F(\alpha) Contain All Zeros of irr(\alpha,F)?

In summary, we have discussed the question of whether all zeros of the monic irreducible polynomial in F[x] with a root of \alpha in E are contained in the extension field F(\alpha). After considering examples such as (x^2+2)(x^2-2) and x^3+3, we have determined that this statement is false.
  • #1
ehrenfest
2,020
1
[SOLVED] extension fields

Homework Statement


Let F be a field and let E be an extension field of F. Let [itex]\alpha[/itex] be an element of E that is algebraic over E. Is it true that all of the zeros of [itex]irr(\alpha,F)[/itex] are contained in the extension field [itex]F(\alpha)[/itex]?
EDIT: I mean algebraic over F

Homework Equations


The Attempt at a Solution

 
Last edited:
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  • #2
Do you mean algebraic over F? And what is irr([itex]\alpha[/itex],F)?
 
  • #3
Yes. See the EDIT. irr(\alpha,F) is just the monic irreducible polynomial in F[x] that alpha is a zero of.
 
  • #4
What are your thoughts about this? Have you tried to come up with a counterexample?
 
  • #5
It is false. Take (x^2+2)(x^2-2) over the rationals. Then zeros of the first factor are imaginary and the zeros of the second factor are real.
 
  • #6
But (x^2+2)(x^2-2) is not irreducible over Q.
 
  • #7
Take x^3+3. It is irreducible by Eisenstein with p = 3. Its zeros are: [itex]\sqrt[3]{3}e^{ik\pi/3}[/itex] where k=0,1,2. Two of those values of k produce imaginary roots.
 
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  • #8
That's better!
 

Related to Extension Fields: Is F(\alpha) Contain All Zeros of irr(\alpha,F)?

1. What are extension fields and why are they important in mathematics?

Extension fields are mathematical constructs that extend the set of numbers used in basic arithmetic operations. They are important because they allow for the solution of equations that cannot be solved using only the basic operations and numbers. They also play a crucial role in areas of mathematics such as algebra and number theory.

2. What is the significance of F(\alpha) in extension fields?

F(\alpha) represents the smallest extension field of the field F that contains both F and the element \alpha. This is important because it allows for the extension of F with new elements that can solve equations that were previously unsolvable in F.

3. What is the relationship between F(\alpha) and the zeros of irr(\alpha,F)?

F(\alpha) contains all the zeros of the minimal polynomial, irr(\alpha,F), which is the polynomial of lowest degree that has \alpha as a root. In other words, F(\alpha) contains all the solutions to irr(\alpha,F) in addition to the elements of F.

4. Can F(\alpha) contain all the zeros of irr(\alpha,F) even if \alpha is not algebraic over F?

No, F(\alpha) can only contain the zeros of irr(\alpha,F) if \alpha is algebraic over F. This means that \alpha must be a root of a polynomial with coefficients in F. If \alpha is not algebraic over F, then F(\alpha) cannot be formed as an extension field.

5. How can extension fields be used in practical applications?

Extension fields have many practical applications in various fields such as engineering, computer science, and physics. They can be used to solve complex equations and model real-world situations. For example, in cryptography, extension fields are used to create secure communication protocols and encryption algorithms.

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