Eragon is heading home with Roran, who tells him that he got a job offer with a miller named Dempton. Eragon is shocked to hear that he's going to take the job. After all, it means leaving Carvahall and his family.
Still, Roran needs the money. For what, you ask? He wants to marry Katrina.
Eragon is sad to hear this, but it makes sense to him. He doesn't think Garrow's going to take it well, though.
When Eragon gets home, he goes out into the woods and confides in the dragon. He's majorly bummed about Roran's leaving. The dragon is a good listener.
Then Eragon decides that the dragon needs a name. (Thanks, Eragon! We were getting tired of calling it "the dragon" all this time.)
He goes through a bunch of the male names he got from Brom, but the dragon's not feeling them.
That's when Eragon realizes that the dragon is a she.
Finally, they settle on "Saphira" for her name. Whew.