example in real/complex:
given 0+i as the element in the extension field (complex), find a minimal polynomial with coefficients in the base field (real). The math will all be done in the extension field (complex) -- only the coefficients of the polynomial as restricted to elements from the base field (real).
One polynomial that has 0+i as a root would be x(x+1)(x^2 +1). This is a 4th degree polynomial. However, it is clear that x(x^2+1), a degree 3 polynomial also has 0+i as a root. Further, (x^2+1) has 0+i as a root. (x - i) is a polynomial of lower order that has 0+i as a root, but it doesn't have all of its coefficients in the base field ( -i is complex). Thus (x^2+1) is a minimal polynomial for 0+i.
This also works for galois fields.
Another related concept is "monic". where (x^2 + 1) and 2x^2 + 2 both have 0+i as a root, "monic" will select the polynomial that has a coefficient of 1 (a concept that exists in any field) as the coefficient of the highest order term.