Can [F: F ∩ E] Differ from 2 in Quadratic Field Extensions?

In summary, the conversation discusses the concept of a quadratic extension, where the field F is contained in K and FE=K. They also consider the lattice of inter-fields between Q and K, which is isomorphic to the subgroup lattice of the group S3. By choosing certain subgroups, they show an example where [F:F intersects E] is not equal to 2, illustrating a non-example for the given question.
  • #1
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if K/E is a quadratic extension and field F is contained in K
such that FE=K and [K:F] is finite,
how do I give a non-example to show
[F: F intersects E] might not be 2?

Thanks a lot!
 
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  • #2
Very EASY!
Do you have Galios theory in your hands?

Well consider [tex]K[/tex] the splitting field of [tex]x^3 - 2[/tex] over [tex]\mathbb Q[/tex].

By Galois theory you should know the lattice of inter-fields between [tex]Q[/tex] and
[tex]K[/tex] is isomorphic to the lattice subgroup of the group [tex]\mathbb S_3[/tex]
of the permutations on 3 elements.

Now this lattice as a unique subgroup on 3 elements and 3 distinct subgroups of 2 elements. Choose two distinct of these and call them [tex]G_1, G_2[/tex].
Let's call [tex]e[/tex] the trivial subgroup (just one element: the identity permutation).

Call [tex]\prime[/tex] the Galois corrispondence and you have the fields

[tex]K = e \prime[/tex]
[tex]E = G_1 \prime[/tex]
[tex]F = G_2\prime[/tex]
and [tex]E \cap F = (G_1\cdot G_2)\prime = \mathbb S_3\prime = \mathbb Q[/tex].

You have [tex][K:E] = [G_1:e] = 2[/tex] and [tex]K/E[/tex] is a quadratic extension
You have [tex][K:F] = [G_2:e] = 2[/tex] and [tex]K/F[/tex] is a finite extension
You have [tex]F\cdot E = (G_2 \cap G_3)\prime = e\prime = K[/tex]
You have [tex][F:E\capF] = [\mathbb S_3:G_2] = 3 \not = 2[/tex].
 
  • #3
Thank you so much!
 
  • #4
You are WELCOME!

Well I also noticed I made a 'print' mistake...

in the last row

I wrote [tex][F:E][/tex] instead of
[tex][F: F \cap E][/tex]

but I guess you noticed the mistake and you got the right meaning.

See you next time!
 

Related to Can [F: F ∩ E] Differ from 2 in Quadratic Field Extensions?

1. What is a field extension problem?

A field extension problem is a mathematical concept that involves extending a smaller field (such as the rational numbers) to a larger field (such as the complex numbers) in order to solve a given problem. This is often done by adding new elements to the smaller field.

2. How do I solve a field extension problem?

Solving a field extension problem typically involves applying algebraic techniques, such as polynomial division and factoring, to find the missing elements needed to complete the extension. It may also involve understanding the structure and properties of the larger field in order to find a suitable solution.

3. What are some real-world applications of field extension problems?

Field extension problems have many practical applications in fields such as cryptography, coding theory, and number theory. They are also used in physics and engineering to model and solve various problems involving complex numbers and other mathematical structures.

4. What are some common challenges when working on a field extension problem?

One common challenge is identifying the correct field to extend to in order to solve the problem. This requires a good understanding of the properties and relationships between different fields. Another challenge can be finding the necessary elements to add to the smaller field, which may require advanced algebraic techniques.

5. Are there any tools or resources available to help with field extension problems?

Yes, there are various mathematical software programs, such as Mathematica and Maple, that can assist with field extension problems by providing computational tools and visualizations. Additionally, there are many online resources and textbooks that provide step-by-step guides and examples for solving these types of problems.

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