Conjugate fields and conjugate subgroups of an automorphism group

In summary, E and D are finite extensions of F and K is the Galois closure of the combination of E and D. E and D are conjugate fields over F if and only if the subgroups G and H, which fix E and D, respectively, are conjugate subgroups of the automorphism group of K over F. Additionally, the Galois closure K is equal to the field fixed by the core of the automorphism groups of D and E over F, but the definition of "core" may not be well-defined in this case since we have not specified a larger group for \text{Aut}(D/F) and \text{Aut}(E/F) to be subgroups of. Perhaps further clarification on the
  • #1
imurme8
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Suppose [itex]E[/itex] and [itex]D[/itex] are both finite extensions of [itex]F[/itex], with [itex]K[/itex] being the Galois closure of [itex]\langle D,E \rangle[/itex] (is this the correct way to say it?) Is it correct that [itex]E[/itex] and [itex]D[/itex] are conjugate fields over [itex]F[/itex] iff [itex]G,H[/itex] are conjugate subgroups, where [itex]G,H\leqslant \text{Aut}(K/F)[/itex] are the subgroups which fix [itex]E,D[/itex]?

I want to claim also that given [itex]E,D[/itex], we have that their Galois closure [itex]K[/itex] is exactly the field fixed by the core of [itex]\text{Aut}(D/F)[/itex] and [itex]\text{Aut}(E/F)[/itex], but I'm not sure if the "core" is well-defined in this case, since we've not defined a group of which [itex]\text{Aut}(D/F)[/itex] and [itex]\text{Aut}(E/F)[/itex] are a subgroup. What do you think?
 
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  • #2
what is the definition of conjugate fields? if you give it precisely maybe you can answer your own question.
 

1. What are conjugate fields in an automorphism group?

Conjugate fields in an automorphism group refer to fields that are isomorphic to each other. This means that they have the same algebraic structure and can be transformed into each other through a mapping that preserves the algebraic operations of addition, subtraction, multiplication, and division.

2. What is the significance of conjugate fields in an automorphism group?

Conjugate fields are important in understanding the structure and properties of an automorphism group. They provide a way to classify and group elements of the group that have similar characteristics and behaviors.

3. How are conjugate fields related to conjugate subgroups in an automorphism group?

In an automorphism group, conjugate fields are associated with conjugate subgroups. This means that for every conjugate field, there exists a corresponding conjugate subgroup that shares the same isomorphism. This allows for a deeper understanding of the relationships between elements within the group.

4. Can conjugate fields and conjugate subgroups be different in size?

Yes, conjugate fields and conjugate subgroups can have different sizes. The size of a conjugate subgroup is determined by the number of elements it contains, while the size of a conjugate field is determined by the degree of its isomorphism. Therefore, they may not always be equal in size.

5. How do conjugate fields and conjugate subgroups affect the properties of an automorphism group?

The existence of conjugate fields and conjugate subgroups in an automorphism group can affect its properties in various ways. For example, they can determine the group's order, its solvability, and its simplicity. Additionally, they can provide insight into the group's structure and the relationships between its elements.

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