Can anyone prove it? I should prove the existence and unique: Let p be an irreducible polynomial of k[x] of degree m. Prove that every element of k[x]/(p) can be represented uniquely by an expression of the form a1x^(m−1) + a2x^(m−2) + · · · + am−1.
- (Existence.) Every element of k[x]/(p) can be represented as described. This just means that each equivalence class of k[x]/(p) contains an element of the kind shown.
- (Uniqeness.) Each equivalence class contains exactly one element of the given kind. As usual assume there are two such elements in a class and show they must be the same. So assume that f = a1x^(m−1) + a2x^(m−2) + · · · + am−1 and g = b1x^(m−1) + b2x^(m−2) + · · · + bm−1 are in the same equivalence class and prove that ai = bi for 1 ≤ i ≤ m − 1, i.e., that f = g in k[x].
Thanks in advance!
Existence : If you have a polynomial with degree $m$ or more, you can divide it by $p$ with remainder. The remainder has the same equivalence class and degree at most $m-1$
Uniqueness : If $f$ and $g$ are polynomials with maximum degree $m-1$, then the difference $f-g$ also has maximum degree $m-1$. Since $f-g$ must be divisible by $p$, it follows $f=g$.