cyclotomic
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Informal working note by Phil dated 1.25.13, said to have been added at the end of Ch 5 of his Galois document. It looks at sets of conjugates of a primitive element α in a finite field, the exponent sets {s, sp, sp^2, ..., sp^(m-1)}, and the name cyclotomic coset. He tries to express the equivalence of s and sp^m using mod (q-1) arithmetic and asks whether sp^m = s mod(q-1). The text is exploratory and unfinished, and the superscripts are partly lost in extraction.
AI-written summary; may contain errors. This description is approximate.
Extracted text (machine-read; may contain errors)
cyclotomic PhL 1.25.13
This has been added into Galois doc at the end of Ch 5.
Consider this conjugate set
{ α, αp, αp, αp , .... αp } αp = αq = α m elements (5.14)
If we compute these sets for every αs, we get these sets (α = primitive so we hit all q-1 nz elements)
{ αs, αsp, αsp, αsp , .... αsp } s ranges from 1 to q-1 since αs are generators
Next element in this set would be
αsp = αs which follows from αp = α
but this is the first element, and that is why we don't include it. It is already included.
In terms of the exponents only, somehow we would say that spm and s are equivalent.
Consider the set of exponents of the conjugates
Cs ≡ { s, sp, sp2, sp3, .... spm-1}
This thing is called a "cyclotomic coset mod p".
How can this equivalence be stated in terms of the mod function?
I think my original hit is just plain wrong. Here is something better
You can write this as
sq = s //equivalent
This would be true in mod q-1 math ?
k 0,1,2,3,4,5,6,7,8
n 0,1,2,3,0,1,2,3,0 8 = 4 = 0 s4 = 4 n = k mod 4
The exponents in this set are
{ s, sp, sp2, sp3, .... spm-1} // next one would be spm which is somehow equivalent to s
This looks close to what my PDF discusses. If I let r = m-1 then this set is
{ s, sp, sp2, sp3, .... spr}
I just said that s lies in 1 to q-1 which also agrees with my PDF. Now can I show that
spr+1 = s mod(q-1) ?
spm = s mod(q-1) ?