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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.

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Just relabel the element $0$ as "apple" in an algebraic closure and its addition/multiplication tables. Then you have a different algebraic closure, because it's a different set. –  blue Aug 5 '14 at 10:43

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up vote 4 down vote accepted

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.

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