On minimum multinomial degrees of algebraic extension fields (Q1121324)

Let K be a field. If \(\alpha\) is an algebraic element in an extension field L of K, the minimum multinomial degree \(d''_{L/K}(\alpha)\) of \(\alpha\) over K, is the least number of non-constant terms in any non- zero polynomial \(f\in L[X]\), such that \(f(\alpha)=0\). For a simple extension L of K, the minimum multinomial degree of L over K is \(d''_{L/K}=\min \{d''_{L/K}(\alpha)| L=K(\alpha)\}\). The authors prove that \(d''_{L/K}<[L:K]\), except in some special cases. They determine some bounds for \(d''_{L/K}\) in the separable case. Finally, extension fields of rational numbers for which \(d''=1\) are studied.