Isomorphic algebraic closures.

Question

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.

Answer 1

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.