We introduce the F-adjoined Gauss map. We use it to express the Gaussian maximum likelihood degree as a product of two invariants. As an application of our product formula, we classify all projective curves of Gaussian maximum likelihood degree 1. We also provide a formula for the generic Gaussian maximum likelihood degree of a projective variety X in terms of its polar classes. The renowned polar class formula for generic Euclidean distance degree is a special case of our formula.