The degree of a polynomial over arbitrary field is not well-defined

Question

I was wondering how a degree of a polynomial can be well-defined when it's representation is not unique. Consider the field $Z_{2}$(a finite field consisting of only two elements) which contains only two elements namely $0$ and $1$ and $1+1=0$ then the polynomials $p_{1}(x)=1$ and $p_{2}(x)=x^2+x+1$ are equal but have distinct representations . As the degree of a polynomial is defined to be the largest exponent in its representation whose coefficient is non-zero then which representation should I consider.

Answer 1

Polynomials are well-defined as infinite sequences with elements in a field (or ring) K such that only finitely many components are nonzero. Here the polynomials are $1 = (1,0,0,\ldots)$ and $x^2+x+1 = (1,1,1,0,0,\ldots)$.

Each polynomial provides a polynomial function over K and in your example as polynomial functions over $\Bbb Z_2$ the functions are equal.