How to prove $t^{23}+1$ irreducible in $F_p$? I have tried to prove that $t^2+1$ is irreducible over $F_3$ by supposing to the contrary $t^2+1=(t+\alpha)(t+\beta)=t^2+(\alpha+\beta)t+\alpha\beta$.
Then, $\alpha+\beta\equiv 0 \pmod 3, \alpha\beta\equiv 1 \pmod 3$. But this leads to $\beta ^2\equiv 2 \pmod 3$ which is not possible by considering cases.
My question would be how do I prove that $t^{23}+1$ (or other similar polynomials like $t^5+1$) is irreducible over $F_p$? Considering cases (linear factors, quadratic factors) seems too tedious to be a feasible method.
Sincere thanks for any help!
[apologies: I have edited the question. The first polynomial should be $t^2+1$ instead of $t^3+1$]

The polynomial $t^3+1$ has the root $-1$ in $F_3$, so it is not irreducible. The same is true of $t^{23}+1$, indeed $t^n+1$ where $n$ is odd. In all cases, the polynomial is divisible by $t-(-1)$, that is, $t+1$.
The same result holds in any field $F$. If $n\gt 1$ is odd, then $t^n+1$ is not irreducible over $F$. The proof is the same.
