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| ``Addition of link to Galois group and of definitions of norm and trace''
by gen_arboit on 2004-09-16 17:37:38 |
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| 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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