Algebraic closure in the context of "Zorn's lemma"

In mathematics, a field F is algebraically closed if every non-constant polynomial with coefficients in F has a root in F. In other words, a field is algebraically closed if the fundamental theorem of algebra holds for it. For example, the field of real numbers is not algebraically closed because the polynomial ${\displaystyle x^{2}+1}$ has no real roots, while the field of complex numbers is algebraically closed.

Every field ${\displaystyle K}$ is contained in an algebraically closed field ${\displaystyle C,}$ and the roots in ${\displaystyle C}$ of the polynomials with coefficients in ${\displaystyle K}$ form an algebraically closed field called an algebraic closure of ${\displaystyle K.}$ Given two algebraic closures of ${\displaystyle K}$ there are isomorphisms between them that fix the elements of ${\displaystyle K.}$

In mathematics, a cubic plane curve , often called simply a cubic is a plane algebraic curve defined by a homogeneous polynomial of degree 3 in three variables ${\displaystyle F(X,Y,Z)}$ or by the corresponding polynomial in two variables ${\displaystyle f(x,y)=F(X,Y,1).}$ Starting from ⁠${\displaystyle f}$⁠, one can recover ⁠${\displaystyle F}$⁠ as ⁠${\displaystyle F(X,Y,Z)=Z^{3}f(X/Z,(Y/Z)}$⁠.

Typically, the coefficients of the polynomial belong to ${\displaystyle \mathbb {R} ,}$ but they may belong to any field ⁠${\displaystyle k}$⁠, in which case, one talks of a cubic defined over ⁠${\displaystyle k}$⁠. The points of the cubic are the points of the projective space of dimension three over the field of the complex numbers (or over an algebraic closure of ⁠${\displaystyle k}$⁠), whose projective coordinates satisfy the equation of the cubic$${\displaystyle F(X,Y,Z)=0.}$$A point at infinity of the cubic is a point such that ⁠${\displaystyle Z=0}$⁠. A real point of the cubic is a point with real coordinates. A point defined over ⁠${\displaystyle k}$⁠ is a point with coordinates in ⁠${\displaystyle k}$⁠.