I really have trouble understanding a task.
We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to show that this is a field "in between" $\mathbb{Q}$ and $\mathbb{R}$. I was explained that this means we have to show:
$$\mathbb{Q} \subseteq \mathbb{Q}[\sqrt{p}] \subseteq \mathbb{R}$$
So that means, the field that we just constructed has to contain every rational number, plus some (not all) of the real numbers. My question is: I found out that $\sqrt{p}$ is always an irrational number $\notin \mathbb{Q}$
If i multiply an irrational number by a a rational number, i will get an irrational number , right?
And if i add a rational number to an irrational number i still get a an irrational number as the result? So doesn't this field we constructed there just contain irrational numbers (This is probably not the case, but why)?
And if i got that right, how can i show that this field contains all rational numbers? (After that i will also have to show that $\mathbb{Q}[\sqrt{p}]$ is a field, but i think i can do that by myself then.)
I am confused.