Showing that a field is isomorphic to a field of vectors

In summary, the conversation discusses a proof that the map defined by \sigma_a(x) = ax is a linear map over the vector space K / F. The conversation also explores the concept of a linear map from a field extension to itself being equivalent to an n x n matrix, and the challenge of showing compatibility between addition and multiplication in this scenario. The speaker is seeking help in finding a more efficient approach to solving this problem.
  • #1
QIsReluctant
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3

Homework Statement


Let [itex]K / F[/itex] be a field extension of degree [itex]n[/itex].

For any [itex]a \in K[/itex] prove that the map defined by [itex]\sigma_a(x) = ax[/itex] is a linear map over the vector space [itex]K / F[/itex]. This part I understand.

Show that [itex]K[/itex] is isomorphic to the subring [itex]F^{n x n}[/itex] of [itex]n x n[/itex] matrices with entries in [itex]F[/itex].

The Attempt at a Solution


I understand that linear map from a field extension to itself = an [itex]n x n[/itex] matrix. The trick is showing that the operations of addition and multiplication are compatible, and I can't seem to do that without a lot of tedious, convoluted calculations, especially since there's not a good closed form for the transformation matrix that would apply to all bases (as far as I know). Is there a better way?
 
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  • #2
I don't think this is supposed to be a hard question, and yet I'm stuck. Any help would be much appreciated.
 

FAQ: Showing that a field is isomorphic to a field of vectors

1. What does it mean for a field to be isomorphic to a field of vectors?

Isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical structures. In the context of fields, isomorphism means that there is a one-to-one correspondence between the elements and operations of two fields. This means that the two fields behave in the same way and have the same algebraic properties.

2. How is isomorphism between fields and fields of vectors established?

To show that a field is isomorphic to a field of vectors, we need to find a bijective function (a function that is both injective and surjective) between the two sets of elements. This function should preserve the operations of addition and multiplication, as well as the identity and inverse elements. In other words, it should map elements from one field to elements in the other field while maintaining their algebraic properties.

3. Why is it important to establish isomorphism between fields and fields of vectors?

Isomorphism allows us to understand and use the properties of one field by studying another field that is isomorphic to it. This can be particularly useful when working with abstract or complex fields, as it provides a more concrete and intuitive way of understanding their operations and properties. Isomorphism also allows us to translate problems and solutions between different fields, making it a powerful tool in mathematics and science.

4. Can any two fields be isomorphic to each other?

No, not all fields are isomorphic to each other. Isomorphism requires a bijective function between the two fields, which means that the two fields must have the same number of elements. Additionally, the operations and properties of the two fields must be compatible and behave in the same way. Therefore, two fields that are not of the same size or do not have similar algebraic properties cannot be isomorphic.

5. How is isomorphism related to vector spaces?

A field of vectors is a vector space, which means that it follows a set of axioms and properties related to vector addition and scalar multiplication. Isomorphism between a field and a field of vectors means that the two fields have the same algebraic properties and behave in the same way, making the field isomorphic to a vector space. This allows us to use vector space concepts and techniques to understand and work with the field, providing a powerful tool for solving problems in mathematics and science.

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