Prydonian Academy

The Prydonian Academy, also known as the Prydon Academy or Prydonian College, was an educational institution on Gallifrey associated with the Prydonian Chapter. Its pupils wore scarlet robes, and were Time Lords by virtue of biology, not education. Amongst other things, the Academy taught them how to instantly calculate "complex temporal induction problems". It had a fortress-like appearance and a population segmented by ability; only a select few were brought to its highest towers to be taught "darker things". (PROSE: Timewyrm: Revelation)

Physically, it was located in Prydos (PROSE: Gallifrey: A Rough Guide) at the base of Mount Cadon, Gallifrey's highest peak. An imposing structure, it was a dark, ebony building (PROSE: Timewyrm: Revelation, Lucifer Rising) at the northern border of "southern Gallifrey". (PROSE: Lucifer Rising)

As Lord President, the Doctor was able to access its so-called "black collection" and steal a portable temporal link. (PROSE: Timewyrm: Revelation)

People associated with the Academy
On one occasion, the Doctor claimed to have attended the Prydonian Academy with the Master, though the Doctor equivocated on the point. (PROSE: First Frontier) At any rate, early on in his life, before becoming a Renegade, the Master was assigned to the Prydonian Academy as a teacher of mathematics and computer science after being recalled to Gallifrey from his field work. (PROSE: CIA File Extracts)

According to the Eighth Doctor, Romana was a Prydonian graduate. (PROSE: The Ancestor Cell)

Before his promotion to Cardinal, Borusa was head of the academy (PROSE: "Academic Progress") and had also been a teacher there. Runcible was a former pupil of his. (TV: The Deadly Assassin)

Neeloc, Taylor Addison, B'arech, Galadina and Karel were students of the Prydonian College. (AUDIO: Insurgency)

Students of the Prydonian Academy were selected and enrolled by the Gallifrey Examining Commission. (PROSE: Gallifrey: A Rough Guide)

Behind the scenes

 * Since "the Academy" typically means the "Time Lord Academy" in works like Interference - Book One and the Gallifrey audio series, and since Paul Cornell barely mentioned the place in Revelation, it's unclear whether the Prydonian Academy is a formal name for the Time Lord Academy, a part of a larger whole, or, indeed, what exactly it is.
 * Insurgency indicates that the institution is a college within the Time Lord Academy.