The Doctor is the protagonist of the British science fiction television programme "Doctor Who" and its adaptions. He is a Time Lord from the planet Gallifrey.
He was the subject of the riddles written in the dust by Arya when she is alone with Eragon in the Empire after the storming of Helgrind. He is likely also referenced by Solembum in Inheritance when he describes the curiosities he has seen throughout his life, including rooms that are bigger on the inside (see below).
Christopher Paolini mentioned the Doctor in his notes at the end of Brisingr, stating "Also, for those who understood the reference to a 'lonely god' when Eragon and Arya are sitting around the campfire, my only excuse is that the Doctor can travel everywhere, even alternate realities. Hey, I'm a fan too!". Another mention to his exsistance is when Eragon is met when leaving Ilirea by Angela and Solembum. Angela is knitting a hat, and Eragon enquires as to what the runes upon it say. She replies with, "Raxacori- Oh never mind. It wouldn't mean anything to you anyway." The word at the start may be Raxacoricofallapatorius, a planet featured in Doctor Who.
Further mentions may include some quotes from Solembum when Eragon is plying the Werecat with questions regarding the advice he had given Eragon in the first book, about the Rock of Kuthian. Solembum said several things that may be about the popular show, including. His speech was as follows:
"There are many strange forces at work in Alagesia, Shadeslayer. I have seen things that defy belief: whirlwinds of light spinning in caverns deep below the ground, men who age backward, stones that speak, and shadows that creep. Rooms bigger on the inside than the outside... Galbatorix is not the only power in the world to be reckoned with, and he may not even be the strongest. Choose carefully, Shadeslayer, and if you choose to go, walk softly."
This could be deduced that the room is the Tardis, and the men who age backwards are/is the Doctor (perhaps this indicates the Doctor, if he has been there, has been to Alagesia on several occasions) though in this case one could also read it as a reference to one of many other figures, a more famous case being that of "Benjamin Button", or the Arthurian sorcerer, Merlin. The Shadows that creep may also be the Vashta Nerada, as seen in the episodes of Doctor Who, Silence in the Library and Forests of the Dead, and the stones that speak could also be a reference to the Weeping Angels, or Lonely Assassins.
His mentioning lends support to the theory that Alagaësia is a parallel version of earth.