Let $F=(f_1,...,f_q)$ be a polynomial dominating map from $mathbb{C}^n$ to $mathbb{C}^q$. We study the quotient ${cal{T}}^1(F)$ of polynomial 1-forms that are exact along the fibres of $F$, by 1-forms of type $dR+sum a_idf_i$, where $R,a_1,...,a_q$ are polynomials. We prove that ${cal{T}}^1(F)$ is always a torsion $mathbb{C}[t_1,...,t_q]$-module. The we determine under which conditions on $F$ we have ${cal{T}}^1(F)=0$. As an application, we study the behaviour of a class of algebraic $(mathbb{C}^p,+)$-actions on $mathbb{C}^n$, and determine in particular when these actions are trivial.