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"Do I look like a bloody rodent?"
While Chileans, who consider the Degu to be a pest rather than a pet, might consider Degus to be rodents, are they?
Wikipedia says "Rodentia is an order of mammals also known as rodents, characterised by two continuously growing incisors in the upper and lower jaws which must be kept short by gnawing"...well, that fits the Degu perfectly, they're famous for their gnawing capability.
Again, according to Wikipedia, "The name comes from the Latin word rodere 'to gnaw'." so at a very basic level, yes, Degus are rodents.
However, here's the mud in the waters of this argument...
"In 1991, a paper submitted to Nature proposed that caviomorphs should be reclassified as a separate order (similar to Lagomorpha), based on an analysis of the amino acid sequences of guinea pigs.[13] This hypothesis was refined in a 1992 paper, which asserted the possibility that caviomorphs may have diverged from myomorphs prior to later divergences of Myomorpha; this would mean caviomorphs, or possibly hystricomorphs, would be moved out of the rodent classification into a separate order.[14] A minority scientific opinion briefly emerged arguing that guinea pigs, degus, and other caviomorphs are not rodents,[15][16] while several papers were put forward in support of rodent monophyly.[17][18][19] Subsequent studies published since 2002, using wider taxon and gene samples, have restored consensus among mammalian biologists that the order Rodentia is monophyletic.[20][21]" (Taken from http://en.wikipedia.org/wiki/Rodent#Monophyly_or_polyphyly.3F)
Degutopia also has a page discussing it, but it hasn't been updated in 6 years (well, no new evidence)...useful for a history of the debate.
So that means Degus, being of the sub-order caviomorph (sub to rodentia), might not be a rodent...they might be under a different classification entirely despite their similarity to rodents....So now the question is if they're not rodents what the hell are they...apart from cute as hell :-)