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 nonnegative integers $n$ the largest degree of an algebraic number whose conjugates span a vector space of dimension $n$ is equal to $2^n n!$. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of $GL_n(Q)$; 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 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.