I am a novice in the field and would appreciate if someone could more thoroughly explain the concept of Cyclotomic Cosets.
The paper http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Calderbank.pdf defines Cyclotomic Cosets as follows: Let n be relatively prime to q. The cyclotomic coset of q modulo n containing i is defined by: $C_i{(iq^j(\mod n)\in\mathbb(Z)_n : j=0,1,...}$.
This is on page 3 of the paper if you want more informtion. Additionally, we have the Cyclotomic Cosets of 15 as: $C_0=(0),$ $C_1=(1,2,4,8)$, $C_3=(3,6,9,12)$, $C_5=(5,10)$, $C_7=(7,11,13,14)$. It is also stated that $C_1=C_2=C_4=C_8$ and similarly for the other entries. The representative entries are {0,1,3,5,7}. I have little understanding as to what all of this entails.
I am also wondering what the significance is to the number of cyclotomic cosets of a number; what does it say about a number; are there any recognizable patterns in the number of cosets, such as for primes?
The co-sets are useful in various isomorphism calculations, such as creating the expansions for the individual values. The idea of isomorphism is that one might scroll through the roots of some equation like $x^n=1$, to work out an algebraic expression for $x$.
A 'co-set' here is a 'parallel set'. The sets here are chain of numbers by way of doubling, modulo 15. So we see the set $C_1 = (1,2,4,8)$, is identical to the set $C_4 = (4,8,1,2)$. One then evaluates all of the sets by doubling until there is no numbers left. The representive entries are simply the smallest number of the set, so $C_7 = C_{11} = C_{13} = C_{14} = (7,11,13,14) $.
The cosets as constructed also represent the numerators in base $j$, when the denominator is $n$, so eg 7/15 = 0.0111, 11/15 = 0.1011, 13/15 = 0.1101, and 14/15 = 0.1110.
The size of the set, is the lowest common multiple of the euler-function of the powers of the primes, the power of 2 to be reduced by 2 if greater than 4. So here, the Euler function of 15 is 8. But the $ \operatorname{lcm}(\phi(3), \phi(5)) = \operatorname{lcm} (2,4) = 4$, so the largest set is four, and all set-lengths divide 4.
For a pure prime, like $17$, there is a set $C_0 = (0)$, and a set $C_1 = (1-16)$. This is because $17$ has a primitive root, which passes through all of the members.
For a prime-power, like $25$, there is still a primitive root that passes through all of the non-multiples of 5, and one for each additional power of 5. So $C_1, C_5, C_0$, of size 20, 4, 1 resp.
In the list {0, 1, 3,5,7}, there are two co-sets of co-primes (1, 7), a co-set of multiples of 3 (3), a co-set of multiples of 5 (5), and a co-set of multiples of 15 (0).