The right to remain silent when arrested or interrogated is guaranteed under

The right to remain silent is protected under the 5th amendment

the Miranda rights. this is why the police read you your Miranda rights when you are arrested.

It was made law that a police officer has to read you your rights when you are arrested because a guy with the last name of Miranda didn't know that he had the right to remain silent when he was taken into custody for an awful crime (I think it was either abduction or rape). He told the police everything because he didn't know he had the right to remain silent. He brought his case to court and won the new law was named the Miranda rights after his last name.