Efficiently solving large-scale multi-field systems arising from regularized fracture models with high-order finite element discretizations remains challenging. This presentation examines how polynomial coarsening can be integrated as an intermediate level within the algebraic multigrid (AMG) hierarchy, creating a semi-algebraic preconditioner (PA-MG). We introduce a polynomial coarsening level that reduces the polynomial degree before standard aggregation-based coarsening. This intermediate step simplifies the system connectivity to resemble that of linear element discretizations, enabling more effective aggregation in the subsequent algebraic multigrid hierarchy. In contrast to re-discretization methods, the semi-algebraic approach employs inherited Galerkin coarsening via element-local transfer operators assembled on a reference element. This requires only element connectivity information, preserving the black-box nature of AMG while leveraging geometric structure. The implementation extends the existing polynomial coarsening functionality in Trilinos MueLu by introducing a stride-based approach. This enables the handling of multiple degrees of freedom per node, allowing for the application of the method to vector-valued problems. We demonstrate the effectiveness of this approach on gradient-enhanced micropolar continuum formulations used for modeling fracture in quasi-brittle materials. These generalized continuum models couple displacements, microrotations, and non-local damage fields, creating a complex multi-field system with strong inter-field coupling. In this context, high-oder finite element discretization is essential for accurately resolving the internal length scale governing damage localization, but poses challenges for traditional algebraic multigrid methods. In the presented simulations, we employ a block preconditioner with inverse approximation using the proposed semi-algebraic PA-MG and compare its performance against a standard block preconditioner using regular AMG. Results show that the semi-algebraic approach significantly improves convergence, nearly halving the required iterations. This efficiency translates to reduced computational time, specially for strong inter-field coupling and complex boundary conditions.