Captain reported a #1 engine roll back at cruise. The Flight Crew attempted a restart; but was unsuccessful and returned to land at the departure airport.

Narrative

While at cruise at FL 340 the number 1 engine started rolling back. We pushed the throttle up but the engine continued to rolled back and eventually failed. We ran the Engine Failure Checklist and then the drift down checklist. The engine indications were normal so we descended to FL 240 to attempt a restart because that was the top of the inflight restart envelope. It seemed to be starting normally except we never got light off so after 30 seconds we terminated the restart attempt. We had sent several messages to Dispatch but ACARS coverage was pretty spotty. We eventually got in touch with Dispatch and agreed ZZZ was a good diversion destination. We had already [requested priority handling]. ATC cleared us directly to a fix on the final approach for the ILS XX. Since I had just finished recurrent last Wednesday I became to pilot flying. Name; the First Officer reviewed all the checklist I had already done and previewed the single engine landing checklist plus we both reviewed the single engine go around if needed. Flew a flaps 15 single engine approach and landing without incident. Exited near the end of the Runway and taxied to a parking spot. Spoke with the Chief Pilot; Maintenance Control; and the crew desk. Overall other than the spotty ACARS coverage I think it went very well.