no further detail.
Generally speaking, where employment is held at will, and is not held pursuant to a collective bargaining agreement or a contract which binds the employer to refrain from terminating employees absent good cause, the employer may terminate the employee for any reason or no reason at all, unlawful discrimination and retaliation excepted.
The corollary to this general principle is that no reason need be told to the terminated employee. They can just tell such an employee that he or she has been terminated.
Delaware is widely known as a jurisdiction which has enacted laws which are very favorable to corporations, which are typically employing entities. I would be quite surprised to discover that the law of Delaware is more favorable to employees than the general rule of law.
However, I don't practice law in Delaware, so don't take what I say here as legal advice. It's just my two cents on the facts you describe in light of general principles of the law of at-will employment. If you need legal advice, please consult a lawyer who holds Delaware licensure. That's not me.
If there's no employment contract specifying otherwise.. absolutely.
It is a surprise to me how common it is for employers not to know about and rely on this right. Legally speaking, it is almost never in the best interests of the employer to state a reason for termination of an employee unless and until a legal or administrative action requires such statement.
My responses to questions on Avvo are never intended as legal advice and must not be relied upon as legal advice. I give legal advice only in the course of an attorney-client relationship. Exchange of information through Avvo's Questions forum does not establish an attorney-client relationship with me. That relationship is established only by individual consultation and execution of a written agreement for legal services.