Irreducible polynomial $t^p-t-a$ in $\mathbb{F}_p$

Question

I'm quite new in polynomials. On lecture lector says that some fact is simple ($f=t^p-t-a$ in $\mathbb{F}_p$, p is prime and $a\neq0$ then f is irreducible), but for me it is not clearly. I don't find similar question.Please, give me hint for this.

Answer 1

These polynomials are called Artin-Schreier polynomials. Their irreducibility over $\mathbb{F}_p$ follows from this Lemma: over $\mathbb{F}_p$, the product of all monic irreducible polynomials with a degree that is a divisor of $k$ is given by $x^{p^k}-x$.

So, in order to check that $x^p-x-a$ is irreducible over $\mathbb{F}_p$, it is enough to show that $$ \gcd(x^{p^k}-x,x^p-x-a)=1 $$ for any $k$ in the range $\{1,\ldots,p-1\}$ (actually, it is enough to show that the above identity holds for any $k$ in the range $\{1,\ldots,\left\lfloor p/2\right\rfloor\}$). In the ring $$\mathbb{F}_p[x]/(x^p-x-a) $$ (which we want to prove to be a field) we have $x^p=x+a$, hence $$ x^{p^2} = (x+a)^p = x^p+a^p = x^p+a = x+2a$$ by Fermat's little theorem and the previous identity is straightforward to prove by induction.