I've just stared learning the Galois Theory so my question might be trivial, but could someone give me an example of two different algebraic closures of the same field? Cause I don't get how they can be different.
There are lots of algebraic closures of $\Bbb R$ in $\Bbb H$ (Hamilton's quaternions. )
For example, you have $\Bbb R[i]$ and $\Bbb R[j]$ and $\Bbb R[k]$. You can adjoin any element that squares to $-1$.
These are all isomorphic to $\Bbb C$, of course.