WIEFERICH PRIMES AND MONOGENIC TRINOMIALS

Abstract

Abstract A prime p $p$ p is called a Wieferich prime if 2 p − 1 ≡ 1 ( mod p 2 ) $2^{p-1}\equiv 1 \pmod {p^2}$ 2 Superscript p minus 1 Baseline identical to 1 left parenthesis mod p squared right parenthesis . A monic polynomial f ( x ) ∈ Z [ x ] $f(x)\in {\mathbb Z}\, [x]$ f left parenthesis x right parenthesis element of double struck upper Z left bracket x right bracket of degree N ≥ 2 $N\ge 2$ upper N greater than or equals 2 is called monogenic if f ( x ) $f(x)$ f left parenthesis x right parenthesis is irreducible over Q ${\mathbb Q}$ double struck upper Q and { 1 , θ , θ 2 , … , θ N <