Interpreting Galois theory: symmetry of i and -i, symmetry of $\sqrt{2}$ and $-\sqrt{2}$

Question

I usually think about Galois theory in this way (interpretation comes from my lecturer's notes from when I was at university):
Go back to the time we only had the reals. Get frustrated with there being no solution to $x^2 + 1 = 0$, and consider adding a new number $i$ to remedy this. Realise that once you extend the field to include the solution $i$, you also add $-i$, another solution, and you have that $i + (-i) = 0$. There is no real difference between the two solutions, they come as a pair and add to $0$. It doesn't matter which you name $i$ and which you name $-i$ (is there even such a thing as singling one out to name it?). I have understood this to be the 2 automorphisms of $\mathbb{R}[t]/\left<t^2+1\right>$ over $\mathbb{R}$.

Is this a consistent interpretation ? I then struggle with another example of extending $\mathbb{Q}$ to include a solution to $x^2 -2 = 0$. This time there are two automorphisms of $\mathbb{Q}[t]/\left<t^2-2\right>$ over $\mathbb{Q}$.
When giving an isomorphism over $\mathbb{Q}$ from $\mathbb{Q}[t]/\left<t^2-2\right>$ to $\mathbb{Q}(\sqrt{2})$, $t$ may equally be mapped to either $\sqrt{2}$ or $-\sqrt{2}$.

Is the situation here analogous ? I ask because while $i$ cannot be compared to reals, only one of $\sqrt{2}$ and $-\sqrt{2}$ satisfy $1<x<2$.

EDIT: By automorphism I mean a field automorphism for a field $F$. This would be a bijective map $\phi$ from $F$ to $F$ which:
_maps the additive identity (0) to the additive identity
_maps the multiplicative identity (1) to the multiplicative identity.
_preserves addition ($\phi(a + b) = \phi(a) + \phi(b)$)
_preserves multiplication ($\phi(ab) = \phi(a)\phi(b)$)

Answer 1

You're definitely on the right track.

But I think it's clearer to consider embeddings instead of automorphisms: (*)

There are two embeddings of $\mathbb{R}[t]/\langle t^2+1\rangle$ into $\mathbb{C}$: they correspond to choosing how to send the class of $t$: to $i$ or $-i$. In this sense, $\mathbb{R}[t]/\langle t^2+1\rangle$ is the universal field that contains $\mathbb{R}$ and a root of $t^2+1$.

In exactly the same way, there are two embeddings of $\mathbb{Q}[t]/\langle t^2-2\rangle$ into $\mathbb{C}$: they correspond to choosing how to send the class of $t$: to $\sqrt2$ or $-\sqrt2$. In this sense, $\mathbb{Q}[t]/\langle t^2-2\rangle$ is the universal field that contains $\mathbb{Q}$ and a root of $t^2-2$.

(*) There is a school of thought that considers field extensions as embeddings. It's quiet profitable.