Editing Algebraically closed field (section)

===The field has no proper algebraic extension===
The field ''F'' is algebraically closed if and only if it has no proper [[algebraic extension]].

If ''F'' has no proper algebraic extension, let ''p''(''x'') be some irreducible polynomial in ''F''[''x'']. Then the [[quotient ring|quotient]] of ''F''[''x''] modulo the [[ideal (ring theory)|ideal]] generated by ''p''(''x'') is an algebraic extension of ''F'' whose [[degree of a field extension|degree]] is equal to the degree of ''p''(''x''). Since it is not a proper extension, its degree is 1 and therefore the degree of ''p''(''x'') is&nbsp;1.

On the other hand, if ''F'' has some proper algebraic extension ''K'', then the [[Minimal polynomial (field theory)|minimal polynomial]] of an element in ''K''&nbsp;\&nbsp;''F'' is irreducible and its degree is greater than&nbsp;1.