Does Being an Automorphism of L Imply Being an Automorphism of K?

  • Thread starter chapani
  • Start date
  • Tags
    Field
For example, let ##F = \mathbb{Q}##, ##L = \mathbb{Q}(\sqrt{2})## and ##K = \mathbb{Q}(\sqrt[4]{2})##. The ##\mathbb{Q}-##automorphism that sends ##\sqrt{2}## to ##-\sqrt{2}## is an ##F-##automorphism of ##L## but not of ##K##. Therefore, (2) does not imply (1).
  • #1
chapani
2
0
let K be an extention field of L and L be an extenstion field of F.

(1) t(a)=a ,for all t e G(K,F)

(2) t(a)=a ,for all t e G(L,F)

where e means "belongs to" , G(K,F) means "set of all automorphisms of K

leaving every element of F fixed and similarly for G(L,F).

i would like to know is (2) implies (1) or (1) implies (2)?

i think (2) implies (1) but not sure.[i have used simple logic no.of automorphisms of K =< no.of auto. of L]

if not then please give answer with counter example,if there no relation between

them then explain with reasion.

thanks in advanced for help me
 
Physics news on Phys.org
  • #2
We have ##F \subseteq L \subseteq K## and an ##F-##automorphism of ##K## is automatically an ##F-##automorphism of ##L##, so (1) implies (2). The opposite is not generally true.
 

FAQ: Does Being an Automorphism of L Imply Being an Automorphism of K?

What is an extension field?

An extension field is a mathematical concept that extends the set of numbers used in a field by adding new elements. These new elements are obtained by adjoining a root of a polynomial equation to the original field.

What is the difference between an extension field and a subfield?

An extension field contains elements that are not present in the original field, while a subfield is a subset of the original field. In other words, an extension field expands the set of numbers used in a field, while a subfield restricts it.

How is an extension field represented?

An extension field is typically represented as F(a), where F is the original field and a is the new element being added to the field. This representation is similar to how complex numbers are represented as a + bi, where a and b are real numbers and i is the imaginary unit.

What is the degree of an extension field?

The degree of an extension field is the number of elements required to generate all the elements in the field. It is also equal to the degree of the polynomial equation used to obtain the new element being added to the field.

How are extension fields used in real life?

Extension fields have various applications in mathematics, physics, and engineering. They are used in coding theory, cryptography, and algebraic geometry, among others. In particular, extension fields play a crucial role in error-correcting codes used in telecommunication systems.

Similar threads

I Finding All Automorphisms of Group
Replies
2
Views
2K
About semidirect product of Lie algebra
Replies
19
Views
2K
A Galois theorem in general algebraic extensions
Replies
6
Views
2K
Valuations and places - decomposition and inertia group
Replies
4
Views
1K
MHB How can we compute the Galois group of a subgroup of a splitting field?
Replies
1
Views
1K
A About universal enveloping algebra
Replies
19
Views
2K
MHB Prove: \(F(a,b)^*=F(a) \cup F(b)\)
Replies
4
Views
2K
MHB Automorphisms of an extension field
Replies
1
Views
2K
Prove relation between generators and automorphisms of Z/nZ*
Replies
4
Views
2K
I Proving inequality about dimension of quotient vector spaces
Replies
0
Views
849

Hot Threads

Recent Insights

Back
Top