What does $x$ $\in$ $\mathbb{Q}$(y) mean?

Question

What does $x$ $\in$ field $\mathbb{Q}$(y) mean? What is $\mathbb{Q}$(10), for instance?

Answer 1

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.

Answer 2

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.