What does $x$ $\in$ $\mathbb{Q}$(y) mean? What does $x$ $\in$ field $\mathbb{Q}$(y) mean?
What is $\mathbb{Q}$(10), for instance?
 A: In general $\mathbb{Q}(x)$ is the smallest extension of $\mathbb{Q}$ which is still a field and which contains $x$. For example, $\mathbb{Q}(\sqrt{2}) = \{ a + b \sqrt{2} : a,b \in \mathbb{Q} \}$. 
Let's check the closure properties for this example. Addition and subtraction are the easy part: you add and subtract "componentwise", getting 
$$(a + b \sqrt{2}) \pm (c+ d \sqrt{2}) = (a \pm c) + (b \pm d) \sqrt{2}.$$
For multiplication, we can expand:
$$(a+b\sqrt{2})(c+d\sqrt{2}) = ac+2bd+(ad+bc)\sqrt{2}.$$
For division, we can rationalize the denominator:
$$\frac{1}{a+b\sqrt{2}} = \frac{a}{a^2-2b^2}-\frac{b}{a^2-2b^2} \sqrt{2}.$$
So this is a field extension. To see that it is the smallest one, note that if $a,b \in \mathbb{Q}$ then we must have $b \sqrt{2} \in \mathbb{Q}(\sqrt{2})$ (for closure under multiplication) and must have $a+b\sqrt{2} \in \mathbb{Q}(\sqrt{2})$ (for closure under addition). So any other extension which contains $\sqrt{2}$, must contain this one, i.e. this is the smallest such extension.
In this sense $\mathbb{Q}(10)$ is just $\mathbb{Q}$. In certain contexts it could be a nontrivial extension, but this is abuse of notation, so some context would be required to be sure.
A: It could be any number of things, but if the $y$ is an integer, then $\mathbb{Q}(y)$ probably means the cyclotomic extension of that order.  So $\mathbb{Q}(10)$ is $\mathbb{Q}(\zeta)$ for $\zeta$ a primitive tenth root of unity.
