Skilled2021-02-16Added 94 answers
The first part of the problem (regarding the degree and the uniqueness of the coset representative) is a consequence of the fact that F[x] is a Euclidean domain with degree as the norm.
By Euclidean algorithm,
Thus, every coset
We have already proved that every coset can be represented by a polynomial b(x) of degree less than the degree of p)(x). Here is the proof of the uniqueness of b(x) (for each coset).
The main point is that for two choices of b(x) and c(x), both of degree
Claim: b(x) is unique.
Coming to the last part, now let F be a finite field.
Claim: R has
Proof: From the previous discussion,number of elements in R = number of distinct cosets in
(a polynomial of degree
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