Roots

Let $F$ be a field or even the ring $\mathbb{Z}/n\mathbb{Z}$. A root of a polynomial $p(x)$ in $F[x]$, is an element $r\in F$ such that $p(x)=0$ in $F$.

As a ``non-example'', we have the following fact which goes back several thousand years ago/

Lemma 2.2.4   $p(x)=x^2-2$ has no roots in $F=\mathbb{Q}$.

proof: (Euclid) Suppose not. Let $r=a/b$ be a root with $a,b$ integers satisfying $gcd(a,b)=1$. Since $a^2/b^2=2$, we have $a^2=2b^2$. In particular, $a$ must be even, say $a=2c$. Then $4c^2=2b^2$, so $2c^2=b^2$. This implies $b$ is even, contradicting out assumption that $gcd(a,b)=1$. $\Box$

Two facts are in sharp contrast to what we are used to for real-valued and complex-valued polynomials:

Fact 1

If $F$ is not a field then it is possible for a polynomial of degree $n$ in $F[x]$ to have more than $n$ roots.

Fact 2

If $F$ is a finite field then it is possible for a (non-zero) polynomial of degree $n$ in $F[x]$ to be zero for every $x\in F$ (i.e., be identically zero on $F$).

For further examples, see the exercises.