Hello, I cannot seem to find in my notes nor in any text a method of finding minimal polynomials. Most minimal polynomials I've encountered are somewhat trivial, but I came across one that I may be tested on that I am unable to solve using a brute-force method:

over

The reason I ask about a method of finding the min. poly. is because I am taking an oral test, and I don't want to be floundering about if I am asked something different than this one in particular.

Anyway, I would appreciate any help. Thanks!

P.s.: That's m_{\sqrt 2 +i}; sorry that the "i" is so tiny.