To find the smallest degree polynomial f(x) with coefficients in Q, such that
The impo
rtant point is that the coefficients should all be in Q. For example,
is not allowed, as its coefficients are not rational numbers.
Now,
So, we expect the minimal polynomial of
to have degree
, a quadrantic extension of a quadrantic extension of
.
We proceed to find this minimal polynomial by repeated squaring (to clear radicals and obtain rational coefficients)
squaring
, rearrange
, square again
, rearrange
Minimal polynomial is