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A raidator in need of a flush will not cause a battery drain.
In most cases a battery drains by itself from not running the tractor. I've had it on a 4020 where the tractor set for a few months and the battery goes almost completely dead. If you think it's more than I'm tellin you here bring it into a John Deere shop to have the wiring checked.
A short in the wiring.
Shorted diodes in the alternator.
dead cell in the battery or something is on pulling power from the battery.
Your alternator could be bad.
Any relay could be the cause.
If no light is on and no relay is stuck then the battery has a dead cell and needs replacing.
I'd recommend you do a battery test and charging system test. The battery would be my first suspect.
Could be a shorted wire, or bad battery.
Good possibility
Early 345's had a pair of wires reversed during manufacture. One had one wire and the other had two; they were either pink or purple with a white stripe. Consult a Deere dealer since they can access the service bulletin on this.