Radar SLAM is attractive for autonomous ground vehicles operating in visually degraded environments, however, scanning radars are noisy, have low scanning rates, and their measurements are challenging to match reliably over long trajectories. This paper presents FD-SLAM, a fast dense radar-inertial SLAM system that extends dense radar-inertial odometry with frequency-domain loop closure and pose graph optimization. The proposed method preserves an image-like structure of scanning radar measurements by using a compact frequency-domain polar descriptor for loop-candidate retrieval and a multi-stage verification pipeline based on temporal filtering, phase-correlation screening, scan-alignment similarity, and geometric consistency checks. Verified loop closures are added as non-sequential constraints in an SE(2) pose graph together with radar-inertial odometry factors. FD-SLAM is evaluated on a publicly available dataset using standard KITTI evaluation metrics. The results show that FD-SLAM improves FD-RIO baseline, achieves competitive performance against current state-of-the-art radar SLAM methods, and provides favorable rotational accuracy across multiple evaluated driving trajectories. Runtime analysis further indicates that the radar-inertial front-end operates above the radar sampling rate on a CPU-only setup, while loop closure detection and graph optimization remain suitable for parallel background execution.