Invariant symplectic forms on number fields

We show that a number field of the form Q(lambda) admits a symplectic form which is invariant under multiplication by lambda if and only if the minimal polynomial of lambda is palindromic of even degree. In particular, if lambda is an algebraic integer, it is forced to be a unit. In the case when the minimal polynomial of lambda is palindromic of degree 2d, we show that there is a d -dimensional space of invariant symplectic forms on Q(lambda).