Find the least prime $p$ such that $x^{22} + x^{21} + \cdots + x + 1$ irreducible over $F_p$.

Question

I unable to proceed. Anyone please help me. Problem from the book Finite Fields and their applications by Lidl and Niederreiter (#2.55 & #2.56).

1. Find the least prime $p$ such that $x^{22} + x^{21} + \cdots + x + 1$ irreducible over $F_p$.

2. Find then the least primes $p$ such that $x^{p-1} + x^{p-2} + \cdots + x + 1$ irreducible over $\mathbb F_2$.

Answer 1

Quoting Theorem $2.47$ from same book (Finite Fields and their application by Lidl and Niederreiter)

Theorem: The cyclotomic field $K^{(n)}$ is a simple algebraic extension of $K$ . Moreover:

i. If $K = \mathbb{Q}$ , then the cyclotomic polynomial $Q_n$ is irreducible over $K$ and $[K^{(n)} : K] = \varphi(n)$ .

ii. If $K = \mathbb{F}_q$ with $\gcd(q, n) = 1$ , then $Q_n$ factors into $\varphi(n)/d$ distinct monic irreducible polynomials in $K[x]$ of the same degree $d$ , $K^{(n)}$ is the splitting field of any such irreducible factor over $K$ , and $[K^{(n)} : K] = d$ , where $d$ is the least positive integer such that $q^d \equiv 1 \mod n$ .

So we have to find the least prime for which $22$ is the least integer $d$ such that $23|(p^{d}-1)$. Thus, the answer is $5$.