To show a field $K$ to be algebraically closed.
122050
Algebraic extension of perfect field in which every polynomial has a root is algebraically closed
81682
Proving the algebraic set corresponding to a polynomial is infinite.
158523
Topology: Proof that a finitely generated cone is closed
158525
Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?
10477
Finding the closure of A=$\mathbb Q \cap [0,1]$.
61429
Is the algebraic closure of a $p$-adic field complete
82824
Affine algebraic subset of $\mathbb{A}_k^4$
134896
Enlightening proof that the algebraic numbers form a field
108156
Algebraic closure of $\mathbb{C}(x)$ is isomorphic to $\mathbb{C}$
139499
The algebraic closure of a finite field and its Galois group
140474
Why is Ext$(-;k) =0$ when $k$ is a field
13606
Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.
132665
Show $p(X)$ (over a field) is irreducible iff $p(X+a)$ is irreducible
123903
If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed?
111338
Affine variety over a field which is not algebraically closed can be written as the zero set of a single polynomial
97004
Algebraic closure not even finitely generated over base field
56171
Is $K(X)$ never algebraically closed?