Is it legal for the police to arrest without a warrant in this situation?

Here's the story...

I live in the state of New Hampshire. A couple months ago my brother got into some trouble, missed his court date, and therefore had a warrant put out for his arrest. He was living with my mother at the time and they weren't getting along, and she didn't want to get into any trouble for housing someone with a warrant so she went to the police station to see if they could go and pick him up. The cop saw that her eyes were red, puffy, and watery and assumed it was because she was drunk, which was not the case. She had been crying for a couple hours, fighting with my brother, so of course her eyes were red and puffy. The cop asked her to do the breathalizer test and she refused. They ended up arresting her even though they had no proof that she had been driving drunk.

At court the other day, the cop admitted that no officer had actually seen her driving and that they had no surveillance footage of her driving either. The judge seems to think that this was not a legal arrest because no one actually saw her driving and she wasn't removed from a vehicle by the police, and that the police would have needed to issue an arrest warrant in order for it to be a legal arrest. If they had actually seen her driving or if she'd actually been pulled over by police, then it would have been a legal arrest without a warrant. But that's not the case, so the judge thinks they would have needed an arrest warrant in order to arrest her, and he's considering throwing out the case.

The judge is giving my mother and the prosecutor a chance to write memorandum.

I'm just wondering if this was a legal arrest, or did they need a warrant?

THANK YOU SO MUCH FOR ANY HELPFUL ANSWERS!