Exploring Subfields of F=F_{p^{18}}: A Lattice Representation

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In summary, the conversation discusses identifying all subfields of a field F=F_{p^{18}}, with p^{18} elements where p is a prime. The suggested approach is to use Lagrange's theorem to find subgroups and subfields with orders that divide p18. Additionally, there is a theorem that states the subfields of Fpn are exactly Fpm where m divides n. This topic is relevant to the Abstract Algebra forum.
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Identify all the subfields of a field [tex]F=F_{p^{18}}[/tex], with [tex]p^{18}[/tex] elements where [tex]p[/tex] is a prime. Draw the lattice of all subfields.

I am allowed to just use a theorem, but I don't know one to use.
 
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Start with Lagrange's theorem (which is for groups, but all fields are additive groups) to get that the order of any subgroup (and hence the order of any subfield) divides p18. This narrows down the possible orders of the fields you're looking for
 
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Also, if you have a subfield K, its group of units K* is a subgroup of F* as well.

Don't you have a theorem that says the subfields of Fpn are exactly Fpm where m divides n?

This stuff belongs in the Abstract Algebra forum.
 

1. What is the purpose of exploring subfields of F=F_{p^{18}}?

The purpose of exploring subfields of F=F_{p^{18}} is to gain a better understanding of finite fields and their properties. This can also help in solving problems related to coding theory, cryptography, and other areas of mathematics and computer science.

2. What is F=F_{p^{18}} and how is it represented as a lattice?

F=F_{p^{18}} is a finite field with p^{18} elements, where p is a prime number. It is represented as a lattice by arranging the elements in a grid-like structure, with each element being a point in the lattice. The lattice structure helps in visualizing the relationships between the elements and understanding the properties of the field.

3. How are subfields of F=F_{p^{18}} identified and explored?

Subfields of F=F_{p^{18}} are identified by finding the divisors of p^{18} and checking if they are prime. These subfields are then explored by studying their properties and relationships with the larger field. This can be done through mathematical proofs, visual representations, and computer simulations.

4. What are some applications of exploring subfields of F=F_{p^{18}}?

The exploration of subfields of F=F_{p^{18}} has several applications in coding theory, cryptography, and other areas of mathematics and computer science. It can help in designing error-correcting codes, creating secure encryption algorithms, and solving complex mathematical problems.

5. What are some challenges in exploring subfields of F=F_{p^{18}}?

One of the main challenges in exploring subfields of F=F_{p^{18}} is the complexity of the field, which can make it difficult to find and understand its properties. Additionally, the large number of elements in the field can make it computationally intensive to explore and analyze. It also requires a strong understanding of abstract algebra and number theory to fully comprehend the subfields and their relationships.

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