is an element of
, which, considered as a vector space over
has basis 1, i. (This means that
; you should know from your lecture that every element of this extension has a minimal polynomial of degree at most 2.)
. We can compute:
So you have 3 vectors in 2-dimensional space, hence there exists a linear combination which zeroes them. This is a question from linear algebra, you should be able to find coefficients a,b,c such that a(1,0)+b(1,1)+c(0,2)=0.
By solving this you get that all such coefficients are multiples of
, hence the minimal polynomial is