Polynomial with no roots over the field $ \mathbb{F}_p $.

Question

How can I show that for any prime $p$ and any $d\ge 2$ there exists a polynomial of degree $d$ in $\mathbb{F}_p[X]$ with no roots? ($\mathbb{F}_p$ is the finite field with $p$ elements).
Thanks in advance for any idea.

Answer 1

Even more than that, there exists an irreducible polynomial of degree $d$. The field $\mathbb{F}_{p^d}$ is a field extension of degree $d$ over $\mathbb{F_p}$. The multiplicative group a finite field is cyclic, so let $\alpha$ be a generator of $\mathbb{F}_{p^d}^{\times}$, and then we have $\mathbb{F}_{p^d}=\mathbb{F}_p(\alpha)$. Since $[\mathbb{F}_p(\alpha):\mathbb{F}_p]=d$ it follows that the minimal polynomial of $\alpha$ is an irreducible polynomial in $\mathbb{F}_p[x]$ of degree $d$.