Was my car searched legally and can I fight my charges in court?

This morning around 10AM ,I rear-ended another vehicle and totaled my car while I was on my way to university. The accident was due to very icy conditions. No one was injured and only one officer responded. He took my name/registration and that of the other person involved in the accident. Everything was fine. I had gotten all my valuables out of the car, they were getting it ready to put on the tow truck, and I was literally walking to my boyfriend's truck to leave the scene when the officer asked me how old I was (I'm 20) and informed me that I was under arrest. I had no idea why and he didn't tell my why until I'd arrived at the station. Apparently at some point he had opened my trunk (this happened as I was leaving the scene and I had never given him permission to search my vehicle, if that matters) and found a zipped cooler with three unopened beers in it. He arrested me, didn't read me my rights, and like I said didn't even inform me why I was being arrested until I arrived at the station even though I asked repeatedly. The car is registered under my dad's name and I share it with him occasionally. He is obviously over twenty-one and I had no knowledge of the alcohol in my trunk (the car was a Honda Civic so you can't see into the trunk unless you open it). The officer did not breathalyze me and had no probable cause to search my vehicle. The vehicle was also not being towed by the police, but by a local small-town towing company. Like I said, he did not ask to search my vehicle nor did I even see him enter my car to pop the trunk open OR search the vehicle because I was leaving the scene. I was arrested, booked, and taken to court and am supposed to appear in front of a DA on Monday. I also have absolutely no criminal record - not even a speeding ticket. Is this something I could fight in court and get away with? I didn't think it was legal for an officer to search a vehicle without permission.