Chapter 5.2, Problem 1E

Confirm the statements made in Example 3 by proving that the following sets are subrings of the ring of all real numbers.

The set of all real numbers of the form $m+n\sqrt{2}$ with $m\in \mathbb{Z}$ and $n\in \mathbb{Z}$.

The set of all real numbers of the form $a+b\sqrt{2}$ with $a$ and $b$ rational numbers.

The set of all real numbers of the form $a+b\sqrt[3]{2}+c\sqrt[3]{4}$, with $a,b$ and $c$ rational numbers.