melodykap

2021-02-15

Let F be a field, and $p\left(x\right)\in F\left[x\right]$ an irreducible polynomial of degreed. Prove that every coset of $F\frac{x}{p}$ can be represented by unique polynomial of degree stroctly less than d. and moreover tha these are all distinct. Prove that if F has q elements, $F\frac{x}{p}$ has $q}^{d$ elements.

2k1enyvp

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.

Let

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.

Proof:

Coming to the last part, now let F be a finite field.

Let

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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