Both student and professor would be guilty of academic dishonesty. This is not, anywhere that I know of, a statutory offense, but academic institutions take it very seriously. The student would probably be regraded as having failed, and would likely be expelled. If the student had already graduated when the bribe was discovered, the university might declare the student's degree void. If the student was expelled, beign admitted to another college might be hard. The professor would quite likely be fired and find it hard to get another job.
However, there are statutes prohibiting commercial bribery in general. These generally prohibit corruptly exchanging a thing of value for an unjust advantage or improper service. According to Wikipedia (linked above), 36 US states have specific laws prohibiting commercial bribery. In CA this is covered by Penal Code 641.3. This provides that Commercial bribery happens when an
- employee corruptly (with intent to injure or defraud)
- solicits or accepts or agrees to accept
- money or anything of value greater than $250
- from someone other than his or her employer
- without the knowledge or consent of the employer
- in return for using his or her position for the benefit of that other person6
Other state laws will probably be roughly similar.
See this news article for some general comments. Such a transaction might well fall under a bribery statute, and could in theory be prosecuted as such. Whether in practice such an offense would be prosecuted is questionable. Given the relatively small amounts of money likely to be involved, a prosecutor might well not choose to bring a case, considering action by the college or university to be sufficient. That is a matter of prosecutorial discretion, and does not affect the legal position.