The Factoriums are the places where Arcane knowledge is dispensed, but what about erudite learning of the non-Arcane but still highly advances sorts?
Starting at the bottom of the sophistication scale and working our way up, there are many systems of learning within the Empire.
The Imperial Primaris is the basic organ of education in the Empire. Adolescents attend the Primaris from the age of nine to twelve. They are taught to read and write Celegian, they are taught to count money and do basic ciphers, they are taught basic Imperial history, and they are taught to venerate the Emperor. The Primaris includes three hours of classwork, three hours of sport, and three hours of study time each day.
After finishing primaris, adolescents usually diverge.
Most practical and hands-on trades and vocations are taught through the Guilds system. There are millions of Guilds in the Empire who build 'stuff'. Roads, buildings, furniture, clothing, carriages, weapons, utensils, household goods, pennants, tapestries, tools, if you can think of it, there is a Guild nearby that makes and sells it. All of that crafstmanship is taught to Apprentices starting at the ages of 13-15. There is a system of Journeyman, Master, High Master, and Artisan in place as more skills and talents are found.
Less hands-on skills such as medicae, scrivener, alchemy, and cipher are taught in the Scholams. Scholams are private or provincial schools of higher learning. Many Scholams offer either a three-year course of training at Imperial expense, followed by three years of Imperial service beginning at age sixteen. The most gifted students can satisfy the Imperial Service requirement by undertaking a three year advanced studies course.