Is minimal polynomial of field of rational numbers and field of real numbers are same? Why? Is minimal polynomial of field of  rational numbers and field of real numbers are same?    Why?
Note that over the field $\mathbb{R}$ of the real numbers irreducible polynomials have degree $\leq2$.
On the other hand over the field $\mathbb{Q}$ of rational numbers you can find irreducible polynomials of any degree, meaning that the degree of irreducible polynomials is not bounded. For instance, if $p$ is a prime number the polynomial
$$
1+X+X^2+\cdots+X^{p-1}
$$
is notoriously irreducible.
This answers your questions because irreducible polynomials are the minimal polynomials of their roots.
Depends.
The minimal polynomial of the imaginary unit $i\in\Bbb C$ is $x^2+1$, which is a polynomial over the rationals and the reals.
But the minimal polynomial of $\sqrt 2\in\Bbb R$ is $x^2-2$ over the rationals and $x-\sqrt 2$ over the reals.