From Aluffi's "Algebra chapter 0" problem 4.10 from chapter 3. Prove that $\mathbb{Q}(\sqrt d)$ is a field and in fact the smallest subfield of $\mathbb{C}$ containing both $\mathbb{Q}$ and $\sqrt d$.
I know how to show it directly by a standard argument (which has been reproduced many times on this site). But the author hints to use field norm. I am a little confused by how exactly are we supposed to use it here.
The only thing I can think of is this:
Let $z\in\mathbb{Q}(\sqrt d)$ and write $z=a+b\sqrt d$. We claim that $\frac{a-b\sqrt d}{a^2-b^2d}$ is the multiplicative inverse of $z$. We only need to show $a^2-b^2d\ne0$. For this we note that $a^2-b^2d=N(z)$ and it is nonzero if $z\ne0$ (which has been proved in the same exercise).
I do not see how we can use norm to show the minimality of $\mathbb{Q}(\sqrt d)$.
This is confusing, since $\mathbb{Q}(\sqrt{d})$ may well be defined to be the smallest subfield of $\mathbb{C}$ containing both $\mathbb{Q}$ and $\sqrt{d}$.
One reasonable interpretation of the question is to suppose that we're looking at the set $\{a+b\sqrt{d}\,\mid\, a,b \in \mathbb{Q}\}$, and calling that $\mathbb{Q}(\sqrt{d})$. Our goal then is to show that this is the smallest subfield containing both $\mathbb{Q}$ and $\sqrt{d}$.
Now, clearly any subfield of $\mathbb{C}$ containing both $\mathbb{Q}$ and $\sqrt{d}$ must contain all numbers of the form $a+b\sqrt{d}$, so the only thing left is to show that this is actually a field. If it is, it has to be the smallest subfield with the required property.
To show that it's a field we need to show that every $a+b\sqrt{d}$ has a multiplicative inverse of the same form, and this is where the norm comes in. Once you show that, you're done.