An example of a field extension where every polynomial has a root but is not algebraically closed
123903
If $E/F$ is algebraic and every $f\in F[X]$ has a root in $E$, why is $E$ algebraically closed?
134619
Galois extension with $X^4-2aX^2+b\in\mathbb Q[X]$
121301
What are the roots of an irreducible $p\in F[X]$ in the field extension $F[X]/\langle p\rangle$?
61428
Show that an algebraically closed field must be infinite.
158525
Given a field, the field of algebraic elements over it in an algebraically closed field extension is again algebraically closed?
157487
Showing a polynomial having at least one integer root under certain conditions has precisely one integer root
131460
Example of finite field extension where root not separable
139499
The algebraic closure of a finite field and its Galois group
113742
Nullstellensatz non-valid for non-algebraically closed fields
111444
Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
109039
Does this "extension property" for polynomial rings satisfy a universal property?
78561
Algebraic field extensions: Why $k(\alpha)=k[\alpha]$.
116955
Showing the existence of a certain kind of extension of a finitely generated algebra
146591
Inseparable polynomial over a non-perfect field
13981
Proof that every polynomial of odd degree has one real root
34740
Do finite algebraically closed fields exist?
19511
Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed?