2 of 5 consistent pronouns

Found mistake in my professors book: Tell her or publish? If so, how?

I just came across quite a big problem for an interpretation my professor puts forward in her recently published book. I am not 100% sure, but I am pretty sure that I am right.

How am I to approach this? Shall I send her an email, explaining everything, or shall I try to publish it as a paper (assuming I am right about the severity of the problem)? I am worried that if I tell her, she might publish a paper herself or will tell me it's not so severe. I am bit paranoid here, I know.

I am also concerned with how to tell her - if I do. Shall I be confident and explain that I think I have found a problem for her interpretation or is this too bold? Or shall I formulate it more like a question? In the latter case, I am worried that she will not realise that I did really see the mistake and am only being polite. I don't want her to think that I pointed to something by asking a question, but that then she really came up with it. I have the feeling it would make more of an impression, if I would point straight to the problem as a problem. Of course, there is then also the possibility of her not liking me for this.

By the way, I am a Master student and will not do my PhD with this professor (or even at the same uni).