Show that $\mathbb{Q}(\sqrt{2})$ is the smallest subfield of $\mathbb{C}$ that contains $\sqrt{2}$ This was an assertion made in our textbook but I have no idea how to show that either statement is true.  Also would like to show that that $\mathbb{Q}(\sqrt{2})$ is strictly larger than $\mathbb{Q}$, which was the second part of the assertion.

$\mathbb{Q}(\sqrt{2})$ is by definition the smallest subfield of $\mathbb{C}$ that contains both $\mathbb{Q}$ and $\sqrt{2}$.  The surprising thing (perhaps) is that any element of $\mathbb{Q}(\sqrt{2})$ can be written in the form $a + b\sqrt{2}$, with $a,b \in \mathbb{Q}$.  To prove this, you have to show the following:


*

*That the set of all elements of the form $a + b\sqrt{2}$, with $a,b \in \mathbb{Q}$ is a field (in other words, that it is closed under addition and multiplication, contains both additive and multiplicative identities, contains additive inverses, and contains multiplicative inverses for all nonzero elements);

*That it contains $\mathbb{Q}$

*That it contains $\sqrt{2}$

