Extention of a field

Question

I am studying about diffrent extentions of a field like F[x], and I have a problem to undrestand how the quotient which is generated by ideal p(x), (p(x) is an irreducible polynomiyal in F[X]) extends F[x]? How does extention and quotient relate to each other? For example how R(x)/ $ x^2+1$ extends R to the field of complex numbers? Thanks for your help.

Answer 1

It goes along these lines: consider the composition of ring homomorphisms: $$F\longrightarrow F[x]\longrightarrow F[x]/(p(x))$$ The first map is the canonical injection, and the second map associates to any polynomial $q(x)$ its congruence class $q(x)+p(x)F[x]$.

As $F$ is a field, and the composition is not the null map, it is injective. If we denote by $\xi $ the class $x+p(x)F[x]$, we have an injection of $F$ into the $F$-algebra $F[\xi]$ generated by $\xi$. Furthermore, as $p(x)$ is irreducible, the ideal $p(x)F [x]$ is maximal, so the quotient $F[\xi]=F[x]/(p(x))$ is a field. This proves $F[\xi] $ is an extension of the field $F$.