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extension field

Defines: 
degree,field extension, base field
Keywords: 
fields, Galois
Synonym: 
extension
Type of Math Object: 
Definition
Major Section: 
Reference
Groups audience: 

Mathematics Subject Classification

12F99 no label found

Comments

Hi,

It would be interesting to add on the possible operations on such a vector space, in the 3rd paragraph.

We know that any element of the Galois group (should have a link here) of a field extension F<E is a linear operator on E over F. Moreover, there are operations which are not field automorphisms, one of which being the multiplication by a fixed element of E. This defines the trace and norm of an element a of E in the following way.

The multiplication map A:E->E defined by Ab=ab is an F-linear operator from E to E. Consider A written in matrix form. The trace of a is defined as the trace of A, which is the sum of the eigenvalues of A. The norm of a is defined as the determinant (or norm) of A, which is the product of the eigenvalues of A. In both cases, the multiplicities are counted.

Thank you,

Genevieve Arboit

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