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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 $\mathbb F_2$.

user26857
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Satya Bagchi
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  • Hint: Try for $x^4 + x^3 + x^2 + x + 1 = \frac{ x^ 5 - 1 } { x - 1 } $ first. – Calvin Lin May 05 '20 at 04:36
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    I have answered this (essentially) here (among other places). In the first question you need $p$ to be a generator $\Bbb{Z}_{23}^$. In the second question you need $2$ to be a generator of $\Bbb{Z}_p^$. Searching for the answer is then straight forward. – Jyrki Lahtonen May 05 '20 at 05:13

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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 $5$.

Jyrki Lahtonen
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Gor Melkumyan
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