Let F be a field with q elements, and let K be an extension of F of degree n. The trace and the norm of an element a € K

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Let F Be A Field With Q Elements And Let K Be An Extension Of F Of Degree N The Trace And The Norm Of An Element A K 1 (29.62 KiB) Viewed 23 times

Let F Be A Field With Q Elements And Let K Be An Extension Of F Of Degree N The Trace And The Norm Of An Element A K 2 (19.38 KiB) Viewed 23 times

Let F Be A Field With Q Elements And Let K Be An Extension Of F Of Degree N The Trace And The Norm Of An Element A K 3 (24.63 KiB) Viewed 23 times

Let F be a field with q elements, and let K be an extension of F of degree n. The trace and the norm of an element a € K are defined as follows: +...+aq”, n-1 Tr(a)= I *(a)=a+a = 204 k=0 n-1 N(a) = [] ¢¹(a) = a · aª …….. · aªª¯¹ · k=0 '99-1
Theorem 5.3. The trace is additive, in fact F-linear, while the norm is multiplicative. The trace and the norm map K onto F.
Exercise 5.4. Show that the kernels of the trace and the norm maps can be de- scribed as follows: {a € K : Tr(a)=0}={b⁹-b:be K}, {a EK: N(a)=1} = {b%/b: bek*}