Linearly independent field homomorphisms.

In summary, the conversation discusses the relationship between linear independence of field homomorphisms and the existence of elements in a field that satisfy a determinant condition. One direction is clear, where the existence of elements with a non-zero determinant implies the homomorphisms are not linearly independent. However, the other direction is not as obvious, where the question asks why linear independence of homomorphisms implies the existence of elements with a non-zero determinant. This concept is known as Dedekind lemma.
  • #1
NoDoubts
20
0
Should be simple, but can't figure out :)

Why is that , for a field K, the linear independence of field homomorphisms g1, ..., gn: K -> K
equivalent to the existence of elements a1, ..., an \in K such that the determinant

det| gi(aj)| != 0 (...so, just like in a case of linear independence of vectors).Thanks!
 
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  • #2
I mean, one direction is clear i.e. if there exist n elements with the det != 0 then homomorphisms are clearly not linearly independent...

I don't see the other direction i.e. given n linearly independent homomorphisms why there exists n elements with the above property?
 
  • #3
guys, need help. this should be easy. pls let me know if anything is unclear.

the statement about different field homomorphisms being linearly independent is called Dedekind lemma. My question is why being linearly independent in this case implies existence of n elements a1, ..., an \in K such that determinant of g_i (a_j) is not zero.

thanks a lot!
 

1. What is the definition of linearly independent field homomorphisms?

Linearly independent field homomorphisms are functions that preserve the operations and structure of a field between two fields. This means that for any two elements in the first field, the homomorphism will map them to two elements in the second field that still satisfy the operations of addition, subtraction, multiplication, and division.

2. How do you determine if a set of field homomorphisms is linearly independent?

A set of field homomorphisms is linearly independent if no linear combination of the homomorphisms can equal the zero homomorphism, where all elements in the first field are mapped to the additive identity in the second field. This means that all the homomorphisms in the set are unique and cannot be obtained from a combination of the others.

3. What is the significance of linearly independent field homomorphisms?

Linearly independent field homomorphisms are important in the study of abstract algebra and the properties of fields. They provide a way to compare and relate different fields, as well as uncovering connections between seemingly unrelated fields.

4. Can a set of linearly independent field homomorphisms also be linearly dependent?

No, a set of linearly independent field homomorphisms cannot also be linearly dependent. This is because the definition of linear independence states that no combination of the homomorphisms can equal the zero homomorphism, meaning that they are all unique and cannot depend on each other.

5. How are linearly independent field homomorphisms used in practical applications?

Linearly independent field homomorphisms have practical applications in areas such as cryptography and coding theory. They are used to construct error-correcting codes and cryptographic algorithms, as well as in the design of efficient data transmission systems.

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