  The field of definition of an affine invariant submanifold M is the smallest
subfield of the reals such that M can be defined in local period coordinates by
linear equations with coefficients in this field. We show that the field of
definition is equal to the intersection of the holonomy fields of translation
surfaces in M, and is a real number field of degree at most the genus. We show
that the projection of the tangent bundle of M to absolute cohomology H^1 is
simple, and give a direct sum decomposition of H^1. Applications include
explicit full measure sets of translation surfaces whose orbit closures are as
large as possible, and evidence for finiteness of algebraically primitive
Teichm\"uller curves. The proofs use recent results of Artur Avila, Alex Eskin,
Maryam Mirzakhani, Amir Mohammadi, and Martin M\"oller.
