Algebraic And Simple Extensions

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In summary, the question posed is whether every algebraic extension of a field is a simple extension. The attempt at a solution involves using the field of algebraic numbers over Q as an example, but the degree of this extension needs to be proven. The tower law is suggested as a method for proving the degree.
  • #1
WannaBe22
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Homework Statement


I'll be delighted to get an answer to the following question:
Does every algebraic extension of a field is a simple extension?

Homework Equations


The Attempt at a Solution


I'm pretty sure that the answer is negative... I was thinking on taking the field of all the algebraic numbers over [tex] Q [/tex] ... this field is obviously an algebraic extension of Q, but how can I prove it isn't simple? (I'm pretty sure that its degree is infinity, but have no idea how to to prove it) ...

Hope you'll be able to help me

Thanks!
 
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  • #2
To prove the degree you can use the tower law. [tex]2^{1/n}[/tex] is obviously in the algebraic numbers for each n, so if A is the set of algebraic numbers

[tex][A:Q]=[A:Q(2^{1/n})][Q(2^{1/n}):Q][/tex]
 
  • #3
Thanks a lot!
 

FAQ: Algebraic And Simple Extensions

What is an algebraic extension?

An algebraic extension is a field extension in which all elements of the extension can be expressed as roots of polynomials with coefficients in the base field. In other words, every element in the extension can be obtained by solving a polynomial equation with coefficients in the base field.

What is a simple extension?

A simple extension is a field extension in which the extension field is generated by a single element over the base field. This means that every element in the extension field can be written as a polynomial of the generating element with coefficients in the base field.

What is the difference between algebraic and simple extensions?

The main difference between algebraic and simple extensions is that a simple extension is a specific type of algebraic extension. While all elements in a simple extension are algebraic over the base field, not all algebraic extensions are simple. In a simple extension, every element can be written as a polynomial of a single element, whereas in an algebraic extension, elements may require multiple generators.

How are algebraic and simple extensions related to finite fields?

Algebraic and simple extensions are commonly used to construct finite fields, which are fields with a finite number of elements. In fact, all finite fields can be constructed as simple extensions of the prime field, which is the smallest field containing 1 and 0. However, not all algebraic or simple extensions result in finite fields.

What is the significance of algebraic and simple extensions in mathematics and science?

Algebraic and simple extensions are important in many areas of mathematics and science, including number theory, algebraic geometry, and cryptography. They provide a way to construct new fields from existing ones, which allows for the study of a wide range of mathematical structures. These extensions also have practical applications, such as in coding theory and data encryption.

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