6.7 An Application to Cyclic and BCH Codes
1.
a. If n = 1 it is clear, if n = 2, 
. In general, 
. In general,
c. If f = a0+ a1x + 
then f′ = a1 + a3x2 + a5x4 + = g(x2) where g = a1+ a3x + . Conversely, f = g(x2) = b0+ b1x2 + b2x4 + clearly implies that f′ = 0 (as 2 = 0 in 
then f′ = a1 + a3x2 + a5x4 + = g(x2) where g = a1+ a3x + . Conversely, f = g(x2) = b0+ b1x2 + b2x4 + clearly implies that f′ = 0 (as 2 = 0 in
3. If n = 2k then
and, by induction, 
; that is 1 − xn = (1 + x)n. So the divisors of 1 − xn are 1, (1 + x), (1 + x)2, . . ., (1 + x)n, and these are a chain under divisibility.
; that is 1 − xn = (1 + x)n. So the divisors of 1 − xn are 1, (1 + x), (1 + x)2, . . ., (1 + x)n, and these are a chain under divisibility.
Conversely, if 1− xn = (1 + x)kpm where p is irreducible ...
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