Polynomials in base \(x\) and the prime-irreducible affinity (Q2420856)

In the paper under review the author is highly motivated by an interesting, but unfortunately less known in literature, criterion for irreducibility of polynomials with integer coefficients, due to \textit{Arthur Cohn} (1894--1940), a student of Issai Schur who was awarded his doctorate from Universität Berlin in 1921, with a thesis entitled ``Über die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise (About the number of roots of an algebraic equation in a circle)'' [Math. Z. 14, 110--148 (1922; JFM 48.0083.01)]. Cohn's irreducibility criterion is often stated as follows: If a prime number \(p\) is expressed in base \(10\) as \(p=a_m10^m+a_{m-1}10^{m-1}+\ldots +a_1 10+a_0\) where \( 0\leq a_i\leq 9\) then the polynomial \(f(x)=a_mx^m+a_{m-1}x^{m-1}+\ldots +a_{1}x+a_0\) is irreducible in \(\mathbb {Z}[x]\). Cohn's irreducibility criterion and its generalization connect primes to irreducible polynomials, and integral bases to the variable \(x\). The author studies this connection by providing some examples and mentioning several related facts. Meanwhile, she proves that there exists a unique base-\(x\) representation of such polynomials that makes the ring \(\mathbb {Z}[x]\) into an ordered domain. Considering this order, she gives a division algorithm for a monic divisor. She also shows that there is a \(1-1\) correspondence between positive rational primes \(p\) and certain infinite sets of irreducible polynomials \(f(x)\) that attain the value \(p\) at sufficiently large \(x\), each generated in finitely many steps from the \(p\)-th cyclotomic polynomial. As a consequence, the author obtains an equivalent to a conjecture due to Viktor Bouniakowsky, known as Bouniakowsky Hypothesis, asserting a generalization of Dirichlet's theorem on primes in arithmetic progressions for higher degree polynomials. The author, finally hopes that the results and techniques in the article will make a new generation of mathematicians take a closer look at the Bouniakowsky Hypothesis with fresh eyes.