Proving Finite Extension is Algebraic & Example of Converse

Q}. However, the converse of the given theorem is not true in general as seen in the example of L = \mathbb{Q} and K = \mathbb{A} (the set of all algebraic numbers). In summary, the theorem states that any finite extension of a field is algebraic, but the converse is not always true as shown by the example of L = \mathbb{Q} and K = \mathbb{A}. Therefore, it is not always possible to prove the converse of this theorem.
  • #1
luciasiti
4
0
Hi everyone
I 'm having difficulty in proving the following theorem
theorem: If L/K ( L is a field extension of K) is a finite extension then it is algebraic. Show, by an example, that the converse of this theorem is not true, in general.
Can you help me to find an example in this case?
Thanks for your help!
 
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  • #2
What algebraic extensions do you know of [itex]\mathbb{Q}[/itex]??
 
  • #3
Let L be the set of rational numbers and K the set of all algebraic numbers.
 
  • #4
micromass said:
What algebraic extensions do you know of [itex]\mathbb{Q}[/itex]??

[itex]\mathbb{Q(\sqrt{2})}[/itex] is an algebraic extension of [itex]\mathbb{Q}[/itex]
 

FAQ: Proving Finite Extension is Algebraic & Example of Converse

What does it mean for finite extension to be algebraic?

Finite extension being algebraic means that every element in the extension can be expressed as a root of a polynomial with coefficients in the base field. In other words, every element is algebraic over the base field.

How do you prove that a finite extension is algebraic?

To prove that a finite extension is algebraic, you can show that every element in the extension satisfies a polynomial equation with coefficients in the base field. This can be done by constructing a minimal polynomial for each element or by using the fact that every finite extension is algebraic.

Can you provide an example of proving finite extension is algebraic?

One example of proving finite extension is algebraic is showing that the extension Q(√2) is algebraic over the base field Q. This can be done by constructing the minimal polynomial x^2 - 2 for the element √2, which shows that √2 is algebraic over Q.

What is the converse of proving finite extension is algebraic?

The converse of proving finite extension is algebraic is proving that every element in the extension is algebraic over the base field. In other words, showing that every element satisfies a polynomial equation with coefficients in the base field.

Why is proving finite extension is algebraic important in mathematics?

Proving finite extension is algebraic is important in mathematics because it allows us to understand the structure and properties of finite extensions. It also has applications in areas such as number theory, algebraic geometry, and cryptography.

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