The conjugate dimension of algebraic numbers

Abstract

We find sharp upper and lower bounds for the degree of an algebraic number in terms of the Q‐dimension of the space spanned by its conjugates. For all but seven non‐negative integers n the largest degree of an algebraic number whose conjugates span a vector space of dimension n is equal to 2nn!. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of GLn(Q); this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when Q is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension Q(ω ℓ) of Q. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.