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

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 $R=F\frac{x}{p}\left(x\right)$, the quotient ring consider any coset $\left(p\left(x\right)\right)+q\right)x\right)\in F\left[x\right]$
By Euclidean algorithm, $\mathrm{\exists }a\left(x\right),b\left(x\right)\in F\left[x\right]$, with $q\left(x\right)=a\left(x\right)p\left(x\right)+b\left(x\right)$ with eirher

Thus, every coset $\left(p\left(x\right)\right)+q\left(x\right)$ is equal to some coset $\left(p\left(x\right)\right)+b\left(x\right)$, with $deg\left(b\left(x\right)\right)$

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 $ the difference is divisible by p(x). Now , a polynomial of lower degree is divisible by p(x) if and only if that polynomial is identically 0.
Claim: b(x) is unique.
Proof: $\left(p\left(x\right)\right)+b\left(x\right)=\left(p\left(x\right)\right)+c\left(x\right)$
$⇔$ $b\left(x\right)-c\left(x\right)$ is diviseble by p(x)
$⇔$ $b\left(x\right)-c\left(x\right)=0$ (as $deg\left(b\left(x\right)-c\left(x\right)\right)$ $⇔$ $b\left(x\right)=c\left(x\right)$
Coming to the last part, now let F be a finite field.
Let $R=F\frac{x}{p}\left(x\right)$,
Claim: R has ${q}^{d}$ elements.
Proof: From the previous discussion,number of elements in R = number of distinct cosets in $F\frac{x}{p\left(x\right)}$ = number of polynomials of degree $
(a polynomial of degree $\sum _{0}^{d-1}{a}_{i}{x}^{i}{a}_{i}\in F$, each ${a}_{i}$ can take q values from the field F)

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