Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix? Do the eigenvalues of a matrix need to be in the same field as the elements of the matrix? I believe they do not since, for instance, I was taught that the trace is the sum of the eigenvalues and this would mean that this is not always the case. What puzzles me is that I have seen people claiming that some real matrix has no eigenvalues simply because the eigenvalues were all complex numbers. Were they right? 

A real matrix can have no eigenvalues in exactly the same way that the real polynomial equation $x^2+1=0$ can have no solutions.
Whether it is true or not depends entirely on who is taking, who they're talking to, and what they are talking about.
