The Complexity of Solving Polynomial Equations by Prime Root Extractions

Abstract

We show that for each algebraic number field Q of finite degree and for each natural number $n = p_1 \cdots p_k ,p_i $ prime, there exists an irreducible polynomial $f_n $ of degree n in $Q[ x ]$ such that $f_n $ is solvable by radicals and $1 + p_1 + p_1 p_2 + \cdots + p_1 p_2 \cdots p_{k - 1} $ extractions of $p_i $th roots are required for obtaining all roots of $f_n $. This generalizes the linear lower bound on the number of circles one has to draw in order to solve a geometric construction problem by ruler and compass and exponentially improves the obvious lower bound which is the number of prime factors of $\varphi ( n )$, the value of Euler’s function on n.