First of all, we must make sure that the patent right is the exclusive right of the applicant, that is to say, in the absence of other agreements, all the proceeds belong to the patentee (that is, the students you mentioned), and the teacher has no right to share them.
However, one question can be drawn from your account:
On the question of the right to apply, you said that the idea was completed with the help of the teacher. Is this kind of help limited to the help of material attributes, or the perfection of technical realization, or the addition of teachers' creative ideas?
If it is the former, the right to apply for a patent is still for students, but if it is the latter, it should be a cooperative invention. According to the patent law, unless otherwise agreed, the right to apply for a patent belongs to the joint entity or individual. In other words, teachers also have the right to stand with students. At this time, teachers have the right to share profits. But the premise is to get the patent right first.
Reply to supplementary question 2:
When you say that the technology for implementing the patent is "known", it means that students' "inventions" can be implemented after ordinary technicians in this field know it. At this time, this kind of "invention and creation" has the "practicality" needed to apply for a patent, and the teacher's implementation is not helpful to the "invention and creation" itself, and it is not a cooperative invention, so teachers cannot share the right to apply.