Three Algebraic Conjugates with Zero Sum: A Graph-Theoretic Approach

Abstract

We give an alternative graph-theoretic proof that 20 is the least degree not divisible by 3 of an algebraic number having three distinct conjugates with zero sum. The proof excludes degree 16 through local constraints on vertex-transitive graphs of valency 6. The underlying graph construction can be studied in arbitrary degree, without the special degree restrictions of our group-theoretic arguments.