The smallest ring containing square root of 2 and rational numbers Can anyone prove why the smallest ring containing $\sqrt{2}$ and rational numbers is comprised of all the numbers of the form $a+b\sqrt{2}$ (with $a,b$ rational)?

Clearly, $Q[\sqrt2]$ is a ring that contains $\sqrt 2$ and all the rationals. Morevoer, every element of $Q[\sqrt2]$ belongs to that smallest ring that contains $\sqrt 2$ and all the rationals. (This is because if a,b are any two rational numbers, then $b\sqrt 2$ belongs to that ring, thus $a+b\sqrt 2$ belongs to that smallest ring)
