Linking two theorems: on algebraic closure and on minimal splitting field

Question

Consider following two statements from same book of Cohn: Basic Algebra.

Proposition 7.3.2: If $k$ is any field and $\mathcal{F}$ is a set of polynomials over $k$, then any two minimal splitting fields for $\mathcal{F}$ over $k$ are isomorphic.

Theorem11.8.3: If $k$ is any field then $k$ has algebraic closure and any two algebraic closures are isomorphic.

My question is, if we take $\mathcal{F}$ to be set of all the monic irreducible polynomials over $k$, then by 7.3.2, doesn't it imply that any two algebraic closures are isomorphic?

Further, just after Theorem 7.3.2, the author gives also the existence of minimal splitting field for $\mathcal{F}$ (it is not visible in link). Then doesn't it imply that algebraic closure of any field exists and is unique up to isomorphism?

In other words, I was thinking that Theorem 7.3.2 and existence of minimal splitting field actually gives proof of theorem 11.8.3 above. Why it was differently stated and proved I didn't understand. Are these two theorems really different?

Answer 1

An algebraic closure must be a splitting field of all polynomials over itself (coefficients from the algebraic closure). So existence and uniqueness of a splitting field of all polynomials over a field $K$ does not trivially imply existence and uniqueness of an algebraic closure of $K$. However, it is not hard to prove that an algebraic extension of an algebraic extension is also an algebraic extension, which indeed bridges the gap, and hence the minimal splitting field of all polynomials over $K$ is indeed the same as the algebraic closure of $K$ (and there is no need to restrict to monic irreducible polynomials).

Answer 2

When you take the splitting field of all irreducible polynomials in a given field $K$ then all those polynomials split in the bigger field, but there is no guarantee that all polynomials in the bigger field split since there are many more new polynomials now. The difficulty with proving an algebraic closure exists is precisely this: it is easy to add the roots of 0any given polynomial, but then you must consider many new polynomials. This is why a simple Zorn's lemma does not work, and one must use the axiom of choice more carefully. In contrast, constructing the splitting field of a polynomial is easy, and does not require choice.