If you got a Trek 420 on the cheap, good job! The 420 was Trek's entry-level touring bike, but Trek's "entry level" bikes back then were still built awfully well.

Along with confirmation via the serial number, you can verify the bike's identity to some extent by checking frame details. I don't know specifically about the 1984 model year, but I've seen 420's that have the Trek name pantographed (engraved) in the seat stay caps at the seat cluster. That would at least confirm it's a Trek. Here are the catalog specs for a 420 frame, some of which you should be able to confirm:

Main Tubes: Tange Mangalloy Double Butted
Fork & Stays: Tange High Tensile
Fork Crown: Tange CCL Semi-Sloping
Bottom Bracket: Nikko with Trek cable guides
Lugs: Nikko Short Point Seamless
Dropouts: Suntour GT Forged
Braze-ons: Top Tube Cable Guides, Water Bottle Mounts, Rear Rack Mounts, Chain Hanger