Rural STEM Education Act This bill directs the National Science Foundation (NSF) to support research regarding STEM (science, technology, engineering, and mathematics, including computer science) education in rural schools. The NSF must award grants to institutions of higher education or nonprofit organizations for (1) research and development to advance innovative approaches to support and sustain high-quality STEM teaching in rural schools, (2) research and development of programming to identify the barriers rural students face in accessing high-quality STEM education, (3) development of innovative solutions to improve the participation and advancement of rural students in grades Pre-K through 12 in STEM studies, and (4) research on online STEM education courses for rural communities. The NSF may establish a pilot program of regional cohorts in rural areas to provide peer support, mentoring, and hands-on research experiences for rural STEM educators of students in grades Pre-K through 12 in order to build an ecosystem of cooperation among educators, researchers, academia, and local industry. The NSF shall enter into an agreement with the National Academy of Sciences under which the National Academy agrees to evaluate aspects of STEM education and workforce development in rural areas. The National Institute of Standards and Technology must award prizes to stimulate research and development of creative technologies in order to deploy affordable and reliable broadband connectivity to unserved rural communities. The Office of Science and Technology Policy must establish a broadband research and development working group to address national research challenges and opportunities for improving broadband access and adoption across the United States.