in algebra one usually proves, that every field $k$ has an algebraic closure $\bar k$ and that all algebraic closures of $k$ are isomorphic. Now I started wondering why people make such a big thing of choosing an algebraic closure, instead of choosing the algebraic closure. I do know that in general for two algebraic closures $\bar k$ and $\bar k'$ of $k$ there are many isomorphisms and that the algebraic closure therefore doesn't have an universal property, but why is this such a big thing?
Thank you for your answer.
We have the theorem that for any two algebraic closures $\overline{k_1}$ and $\overline{k_2}$ of $k$ there exists an isomorphism $\overline{k_1}\cong \overline{k_2}$ over $k$.
Here it is important that the isomorphism in the theorem is nearly always highly non-unique (it can be composed with any k-automorphism of $\overline{k_2}$, of which there are many in general). Thus, one should never write $\overline{k_1}= \overline{k_2}$; one should always keep track of the choice of isomorphism.
In particular, one should always speak of an algebraic closure rather than the algebraic closure; there is no “preferred” algebraic closure except in cases when there are no non-trivial automorphisms over $k$.
So this is not a "big thing", but worth to mention it, I guess.