Fields Suppose F=Q, K=Q(sqrt (3)) and E=Q(sqrt(12)). Find the minimum polynomials h(x) for sqrt(3) and g(x) for sqrt(12) over Q.

Solution: The minimal polynomials are

h(x) = x^2 - 3

g(x)= x^2 - 12, as they are the least degree polynomials over rationals having roots as sqrt(3) and sqrt(12) as roots respectively.

Now, the only roots of h(x) are sqrt(3) and -sqrt(3).

And the roots of g(x) are sqrt(12) And - sqft(12).

Thus, the roots of g(x) and h(x) are not common.

(Please feel free to ask any doubts regarding the solution and please rate positively.

Kind Regards.)