Abstract Algebra: Proving E=F(a^p)

In summary, in this conversation, the speaker is discussing the relationship between a field of characteristic p and its extension E, which is generated by the addition of an element a that is separable over F. The speaker is attempting to prove that E is equal to F(a^p) by considering cases where E is either finite or infinite and using the concept of the Frobenius endomorphism. However, it is pointed out that the isomorphism between E and Z mod p^k can only exist in certain conditions and cannot be generalized.
  • #1
dabien
4
0

Homework Statement


Let F be a field of characteristic p>0 and let E = F(a) where a is separable over F. Prove that E=F(a^p).


Homework Equations





The Attempt at a Solution


I know that maybe show how mod F(a) = mod F(a^p) or something around there.
 
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  • #2
So, since the characteristic is p, a prime E is generated by some field F adjoined some set of elements, and for each minimal polynomial associated to these elements there are no duplicate roots.

Now, there are two cases to consider E is finite or E is infinite. If E is finite then you're done, since E is iso to Z mod p^k. If E is infinite (for example Z4(u) where u is some transcendental element) then you have some work to do. Consider what frobenius endomorphism
http://en.wikipedia.org/wiki/Frobenius_endomorphism
Tells about the minimal polynomial of a.
 
  • #3
icantadd said:
E is iso to Z mod p^k.
No it's not. Such an isomorphism can only exist when k=1 and E is a field whose cardinality is p.
 

Related to Abstract Algebra: Proving E=F(a^p)

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures, such as groups, rings, and fields, using abstract concepts instead of specific numbers or objects.

2. What does it mean to prove E=F(a^p) in abstract algebra?

Proving E=F(a^p) means showing that two algebraic structures, E and F, are isomorphic, meaning they have the same underlying structure and operations. In this case, a^p is an element of both E and F, and the isomorphism preserves the relationship between a^p and the other elements of E and F.

3. Why is proving E=F(a^p) important in abstract algebra?

Proving E=F(a^p) is important in abstract algebra because it allows us to understand the structure and properties of a given algebraic system by studying a simpler isomorphic system. This can lead to deeper insights and generalizations about algebraic structures.

4. What are some common techniques for proving E=F(a^p)?

Some common techniques for proving E=F(a^p) in abstract algebra include constructing an explicit isomorphism between the two structures, showing that the structures satisfy the same axioms or properties, and using theorems and results from other areas of mathematics, such as group theory or field theory.

5. How does proving E=F(a^p) relate to other areas of mathematics?

Proving E=F(a^p) is related to other areas of mathematics, such as group theory and field theory, because it involves studying the structure and properties of algebraic systems. Additionally, isomorphisms between algebraic structures can help us understand relationships between different mathematical objects and theories.

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