How to define an algebraically closed field A field $K$ is algebraically closed if every non-constant polynomial $f \in K[x]$ has a root in $K$, i.e. there exists $a \in K$ such that $f(a) = 0$. Some facts I've noticed:


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*$\mathbb{C}$ is algebraically closed (fundamental theorem of algebra).

*$\mathbb{R}$ is not, since $f(x) = x^2 + 1$ has no root in $\mathbb{R}$.

*$\mathbb{Q}$ is not, since $f(x) = x^2 - 2$ has no root in $\mathbb{Q}$.

*No finite field is algebraically closed.


So it seems that to determine if something is not algebraically closed, that you just find one example of a polynomial in $f \in K[x]$ where $f$ has no root. This is just me speculating based on those few examples.
So this question boils down to:


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*How to find an example polynomial $f \in K[x]$ where $f$ has no root. Wondering if there are general techniques or specific algorithms. Or do people use alternatives approaches such as these to determine it.

*How to prove that there is not a case of "no root". That seems hard. Wondering if there is a general approach. That is, an approach to proving (roughly showing) that you can't find any $f \in K[x]$ such that $f$ doesn't have a root.
So given some data (some vectors, a field, a ring, etc.), how to morph it into an algebraically closed field. How to know that it is actually closed.
I am looking for a way to figure out if a field associated with a given polynomial ring is algebraically closed, assuming the polynomial ring and fields might not be consisting of numbers and/or matrices, but would have a "zero element" and related requirements. Not looking necessarily for a rigorous proof, just general techniques / high-level overview.
 A: There are roughly three main ways to obtain an algebraically closed field and prove it is algebraically closed.
The first is by some sort of analytic completeness.  You construct a field using methods of analysis, and then prove that any polynomial has a root ultimately because you can in some sense "approximate" roots and your field is closed under taking limits of such approximations.  The most famous and by far the most important example of this is of course $\mathbb{C}$.
The second is to just construct your field to be algebraically closed by brute force.  Start with a field $K$, and just keep adjoining roots of polynomials to $K$ until every polynomial has a root, and voila, you have an algebraically closed field $\overline{K}$ (called an algebraic closure of $K$)!  This is actually how most algebraically closed fields arise, other than those that are closely related to $\mathbb{C}$.  Note that this process of continually adjoining roots usually takes infinitely long (sometimes uncountably long) since there are infinitely many polynomials whose roots you need to adjoin, and as a result is often far from constructive: you don't actually have any tangible description of what $\overline{K}$ is.  In particular, proving the existence of such an algebraic closure in general requires the axiom of choice.
The third method is by building off of algebraically closed fields which you already have to build related fields which are algebraically closed because the field you started from was algebraically closed.  For instance, starting from $\mathbb{C}$, you can take the subfield of all elements which are roots of polynomials with integer coefficients, and this subfield (known as the algebraic numbers) is algebraically closed because $\mathbb{C}$ is and it is closed within $\mathbb{C}$ under taking roots of polynomials.
