Exploring Degree Odd Polynomials and Extension Fields of K

So if we let u=a^2, then we can show that K(u)=K(a^2).In summary, we can use the theorem [F:K]=[F:K(u)][K(u):K] to show that K(a^2) is contained in K(a) and [K(u):K(a^2)]=1 or 2. Additionally, based on the fact that the minimal polynomial has an odd degree, we can show that K(u)=K(a^2) by letting u=a^2.
  • #1
kathrynag
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1.Let F be an extension field of K and let u be in F. Show that K(a^2)contained in K(a) and [K(u):K(a^2)]=1 or 2.

2.Let F be an extension field of K and let a be in F be algebraic over K with minimal polynomial m(x). Show that if degm(x) is odd then K(u)=K(a^2).



1. I was thinking of doing something like [K(u):K(a)][K(a):K(a^2)]

2. algebraic so there exists a polynomial m(x) such that m(a)=0.
Just not sure how to work in the degree being odd.
 
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  • #2
Is this a good way to start or should I try something different?
 
  • #3
2. I was thinking of somehow using a theorem stating [F:K]=[F:K(u)][K(u):K]
based on the deg m(x) being odd, I would say deg m(x)=2n+1
 

1. What are odd polynomials and how are they different from even polynomials?

Odd polynomials are mathematical expressions that contain only odd powers of the variable. This means that the exponents on all terms in the polynomial are odd numbers, such as x, x^3, x^5, etc. In contrast, even polynomials contain only even powers of the variable, such as x^2, x^4, x^6, etc. The main difference between the two is that odd polynomials are symmetric about the origin, while even polynomials are symmetric about the y-axis.

2. What is the degree of an odd polynomial and how can it be determined?

The degree of a polynomial is the highest power of the variable present in the expression. For an odd polynomial, this will be the exponent on the largest odd power of the variable. For example, in the polynomial x^5 + 3x^3 + 2, the degree is 5 because it is the largest odd power of x. To determine the degree of an odd polynomial, simply find the exponent on the largest odd power of the variable.

3. How are extension fields of K related to exploring degree odd polynomials?

Extension fields of K are mathematical structures that are used to extend the set of numbers that a polynomial can have as solutions. This is particularly useful when working with odd polynomials, as their solutions may not always be found in the original field K. By exploring extension fields of K, we can find additional solutions to these polynomials and gain a better understanding of their properties.

4. Can all odd polynomials be factored into linear factors over a given field K?

No, not all odd polynomials can be factored into linear factors over a given field K. This is because some odd polynomials may have complex or irrational solutions that are not included in the set of numbers in field K. However, using extension fields of K, we may be able to factor these polynomials into linear factors over a larger set of numbers.

5. How can exploring degree odd polynomials and extension fields of K be applied in real-world situations?

Exploring degree odd polynomials and extension fields of K can have various practical applications, particularly in the fields of engineering, physics, and cryptography. For example, in engineering, odd polynomials are often used to model nonlinear systems and their solutions can be found using extension fields of K. In physics, odd polynomials can represent the motion of objects under the influence of friction and other forces. In cryptography, exploring these concepts can help in the development of secure encryption algorithms.

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