Degree of extension invariant upto isomorphism?

In summary, if $K$ is a field and $F_1$ and $F_2$ are subfields of $K$ that are isomorphic as fields, and $[K:F_1]$ is finite and equal to $n$, then it is necessary that $[K:F_2]$ is also finite and equal to $n$. This can be proven if the isomorphism between $F_1$ and $F_2$ can be extended to an automorphism of $K$. Otherwise, a counterexample can be constructed, such as with $K = \mathbb{Q}(x)$, $F_1 = \mathbb{Q}(x)$, and $F_2 =
  • #1
caffeinemachine
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Let $K$ be a field and $F_1$ and $F_2$ be subfields of $K$. Assume that $F_1$ and $F_2$ are isomorphic as fields. Further assume that $[K:F_1]$ is finite and is equal to $n$.

Is it necessary that $[K:F_2]$ is finite and is equal to $n$??
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I have not found this question in a book so I don't know the answer to the above question. I could not construct a counterexample.
If the isomorphism between $F_1$ and $F_2$ can be extended to an automorphism of $K$ (this probably required additional hypothesis) then the result can be proved in the affirmative.
 
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  • #2
Here is an idea:

Let $F = {\mathbb Q}$ (to fix ideas). Consider $F_1 = F(x)$ and $F_2 = F(x^2)$, and finally $K = F(x)$. The field $F_1$ should be isomorphic to $F_2$ since $x^2$ is trascendental over $F$.

Clearly $K/F_1$ has degree 1. However ;).
 
  • #3
PaulRS said:
Here is an idea:

Let $F = {\mathbb Q}$ (to fix ideas). Consider $F_1 = F(x)$ and $F_2 = F(x^2)$, and finally $K = F(x)$. The field $F_1$ should be isomorphic to $F_2$ since $x^2$ is trascendental over $F$.

Clearly $K/F_1$ has degree 1. However ;).
So $K/F_2$ doesn't have degree $1$ simply because $1$ and $x$ in $K$ are LI over $F_2$. Is that right?
 
  • #4
caffeinemachine said:
So $K/F_2$ doesn't have degree $1$ simply because $1$ and $x$ in $K$ are LI over $F_2$. Is that right?

Right. Otherwise $x\in F_2$ and so $x = f(x^2) / g(x^2)$ for some polynomials $f$ and $g$ ($g\neq 0$), which you can check is nonsense.
 
  • #5
PaulRS said:
Right. Otherwise $x\in F_2$ and so $x = f(x^2) / g(x^2)$ for some polynomials $f$ and $g$ ($g\neq 0$), which you can check is nonsense.
That's great Paul.

I have one more question.

I think that $[K:F_2]$ is finite.
My Argument:
Clearly $K=F_2(x)$. Now define a polynomial $p(y)=x^2y^3-x^4y$ in $F_2[y]$. Clearly $p(x)=0$. Thus $[K:F_2]$ is finite.

Is this okay?
 
  • #6
caffeinemachine said:
That's great Paul.

I have one more question.

I think that $[K:F_2]$ is finite.
My Argument:
Clearly $K=F_2(x)$. Now define a polynomial $p(y)=x^2y^3-x^4y$ in $F_2[y]$. Clearly $p(x)=0$. Thus $[K:F_2]$ is finite.

Is this okay?

Correct. :)

There is a simpler polynomial $p(y)\in F_2[y]$ such that $p(x) = 0$, can you find it?
 
  • #7
PaulRS said:
Correct. :)

There is a simpler polynomial $p(y)\in F_2[y]$ such that $p(x) = 0$, can you find it?
$p(y)=y^2-x^2$. That's just awesome!
 

FAQ: Degree of extension invariant upto isomorphism?

What is a degree of extension invariant?

A degree of extension invariant is a mathematical concept used in the study of field extensions. It measures the "size" of a field extension by counting the number of intermediate fields between the extension and the base field.

What does "upto isomorphism" mean in this context?

"Upto isomorphism" means that two field extensions are considered equivalent if they are isomorphic, or have the same structure. This allows for a more general definition of degree of extension invariant.

How is degree of extension invariant calculated?

Degree of extension invariant is calculated by counting the number of intermediate fields between the extension and the base field. It is also equal to the degree of the extension if the base field is the prime field.

Why is degree of extension invariant important?

Degree of extension invariant is important because it allows for the comparison of field extensions and the classification of fields. It is also used in various algebraic and number theoretic problems.

What are some applications of degree of extension invariant?

Degree of extension invariant has applications in algebraic geometry, algebraic number theory, and Galois theory. It is also used in cryptography, coding theory, and other areas of mathematics.

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