Definition Of Polynomial Equation

Question

I need some clarification on the definition of polynomial equation.

Let $f(x) = \displaystyle\sum_{i=1}^n f_i \,x^i ,\; g(x) =\displaystyle\sum_{i=1}^m g_i \, x^i\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field. We know that $f = g \,$ if $m=n$ and $\forall i, f_i = g_i$. This also normally coincides with the definition of the equality of functions, where $f = g$ if $ f(x)=g(x)$ for all $x$.

However consider the following polynomial over $\mathbb{Z}_2[x]$: $f(x) = x^2 + x + 1,\; g(x) = 1$. These two polynomials obviously do not have equal coefficients however $f(x)=g(x)$ for $x = 0, 1$.

So are these two polynomials equal or different?

Answer 1

You are misunderstanding polynomials and polynomial functions.

Let $p$ be a prime number and let define $f=x^p-x\in(\mathbb{Z}/p\mathbb{Z})[x]$. Using little Fermat's theorem, one has: $$\forall x\in\mathbb{Z}/p\mathbb{Z},f(x)=0.$$ So $f$ as a polynomial function is equal to $0$ over $\mathbb{Z}/p\mathbb{Z}$, but $f$ is not the zero polynomial of $(\mathbb{Z}/p\mathbb{Z})[x]$.

To go back to your example, $x^2+x+1$ and $1$ are equal as polynomial functions over $\mathbb{Z}/2\mathbb{Z}$ but are different polynomials of $(\mathbb{Z}/2\mathbb{Z})[x]$.

What is interesting to notice is that one has the following:

Proposition. Let $R$ be an infinite integral domain and $f,g\in R[x]$. If for all $x\in R,f(x)=g(x)$, then: $$f=g\textrm{ in }R[x].$$

Proof. Since $f-g$ has an infinite number of roots in the fraction field of $R$, $f-g=0$ as a polynomial. $\Box$

This proposition basically tells us that polynomials and polynomials functions behave in the same way over a infinite integral domain.