So my first university class in my first year is a linear algebra course for second year and the teacher was explaining fields to us then at the end of the class he asked us this:
$$\begin{align*}F &= \mathbb{Q}(\sqrt5) \subset \mathbb{R}\\ &= \{a+b\sqrt5 : a,b \in \Bbb Q\}\end{align*}$$
Show that the inverse exists...
I'm like 100% sure this is a really easy question although i haven't done math in months and haven't done much proofs either so cut me some slack please.
You want to show that (presumably multiplicative) inverses exist for general members $a+b\sqrt5\in F$. Consider: $$(a+b\sqrt5)(a-b\sqrt5)=a^2-5b^2$$
Do you see now?