We have solved an important mathematic problem that has not been solved over 10 years. I want to publish a paper about this topic soon, but my supervisor impede me, and he is reasoning that the quality of paper is not good enough. I am concerned that he may publish a paper about the mathematic solution, without mentioning me as an author, to claim that he is the first person to solve the mathematic problem. Is this action legal?
Collaborating with someone you don't trust is no fun. I do not speak from direct experience -- I have certainly trusted and remained on positive terms with all of my collaborators. Yet even under these favorable conditions there are (most) often some conflicts with one's collaborators: e.g. (i) they are dragging their feet / pushing you forward impatiently, (ii) including sloppy and inaccurate / pedantically correct and overly lengthy text, (iii) pushing for publication of a modest partial result / never being satisfied with what has already been obtained, (iv) shooting way too low / way too high in the choice of journal...and so forth. If I thought that my collaborators truly did not have my best interests at heart...yikes. This goes doubly when there is a significant differential of power / seniority among collaborators: there can be a fine line between giving your junior collaborator the benefit of your additional experience and insisting that they do things your way.
I'm very sorry to hear that you are worried that your advisor may try to steal your joint work. If your question is really whether it is in any way ethically acceptable or appropriate for him to do so, the answer is easy: of course not. However, that you ask shows either great inexperience / mild paranoia on your part or is a sign that the collaboration is dangerously in the "no fun" range. I looked at your profile and saw that you are a PhD student at a quite reputable research university in the US. Moreover based on your username I suspect that you are either one of two students with that name in a certain department (not the math department) at that university, each with the same advisor, a young faculty member with an impressive CV. Obviously I can't promise anything, but in all of my experience a faculty member with a CAREER grant is not someone who would want or need to steal their student's work: not even close, and not on their worst day. (As an aside, the fact that I suspect that I have identified your advisor whom you are one step away from accusing of stealing your joint work is not great, and you should think about being a bit more circumspect.)
I advise you to concentrate on improving the interpersonal aspects of your collaboration: try to understand your advisor better and vice versa, and try to shore up trust. If you really can't do that, it's time to find a new advisor. Though I don't know the situation, based on the fact that you've given no convincing evidence for your suspicions and some other clues, my guess is that you are rather new to research (at least of this kind or at this level) and that as you "learn the ropes" your suspicions will be assuaged. If you have made a significant mathematical (note: not "mathematic"!) breakthrough, then indeed you should proceed carefully and patiently and not rush into publication. If the paper is not well written -- according to standards that are difficult to meet or even fully grasp for most graduate students -- that can affect its reception and jeopardize its suitability for publication in a top journal.
It is much more likely that this paper really is of low quality, and your advisor is trying to help you. Listen to what they have to say, and make these changes they suggest. You're supervisor has very little to gain (and quite a bit to lose) by stealing your work. I'm assuming you're a Ph.D student, and your advisor is the one stopping you.
While it might not be illegal, stealing a student's work would be very unethical, and not in the prof's best interest. The prof would be an author. He/she will already get some credit for it anyway. No prof is going to risk his or her (tenured?) position just to be sole author.
If your advisor is truly trying to steal your work, then you'll have email trails and first drafts to prove you came up with the idea.
As a student, it is sometimes difficult to understand the subtle nuances of presenting a paper effectively. What you feel is a great paper with groundbreaking results may not seem to your advisor (with his higher levels of maturity and experience) to be exactly of publishable quality. Generally speaking, advisors in any field would act in the best interest of the student. Moreover, in this case, since he himself is a co-author, he would also have his own interest in mind. A senior professor would definitely have high standards of quality and would not want to be associated with a paper unless he is assured of its quality. I think you should just relax and trust your advisor. I'm quite sure that a few years down the line, you'll be glad that you did not rush into publication and followed your supervisor's advice. Possibly, if you work on polishing your paper some more, it will stand better chances of being published in a top-tier journal.
Just to complement the already good answers provided, I'm fairly surprised no one mentioned this possibility: in mathematics, is fairly frequent that one thinks has proven something, when an error is lurking somewhere in the proof.
What that means, practically, is that your advisor might feel he needs more time to make sure the proof is indeed correct.
It depends on what country you are working in. If you are in the US or another country with a strong rule of law and high academic ethics standards, then it is extremely unlikely that your supervisor will steal your research. If you are in a developing country with dysfunctional legal system, then yes it is possible. I worked in a post soviet country for a brief period of time and my grandparents were scientists in the Soviet Union. Theft of intellectual property there was common place. My grandfather's colleague republished my grandfather's thesis in English and got a fellowship in the US with it. Not much my grandpa could do from under the iron curtain.
The first thing that happens before someone steals your idea is that she becomes convinced that it is, in fact, HER idea. Of course you are a young scientist and your work is not of the high quality yet and you need to improve it. Just make sure that when you are working on it, your supervisor is aware that it is YOU who are improving it. With that being said you should work on it actively and don't forget about it until someone else finishes it up.
I agree with the already good answers. If this situation really bothers you, a possible action could be to write a paper draft with your name on it and timestamp it. You most likely wont need it, but you have a way to protect your work.
If you are extremely worried that you work not receive credit. Look into your Universities Intellectual Properties contact. As a student you have agreed to the universities definition/beliefs on Intellectual Property/Original Authorship. If you used any advanced equipment provided by the University to solve, or invent, most universities take some credit in discoveries. If it was for a class project, then most likely your university receives some credit, if you presented this as a project paper, your university might get some credit. If your really concerned, take your scenario, and apply it to your Universities rules regarding said issue. Only you know if the advisor helped, or if this was a project paper, or a team effort. Also their are many things to show this solution was yours if he tries to publish. First revamp your draft, then shooting a email to the adviser asking he/she to give you feed back on the paper. Make it clear that this is your discovery, that He/she had nothing to do with it. Anyone that sees said email and attachment need to understand that your the original author just by skimming over it. Most importantly, make it subtle as well. Don't make it blatantly obvious as to what your goal is. Make sure the advisor can't pull you aside to task about the email. You want a response. This will be highly incriminating should original authorship belong to you alone. Follow this method with a backup. I would suggest taking this paper, and mailing it to a close family member. Ensure they do not break the seal when it arrives. It will receive a date stamp via the U.S. postal service. Then to prove it you have said mail brought forward and opened when needed. You can also open a date deposite box with most banks for around $15/month. Put a copy of your paper in there and don't go back. You can prove the document was written before any other published papers. The bank will have a log of when you first when in there upon setting up your box. Only accessing the box when you first created it shows that what ever you out there has been there since said date.