Of all of the sections in this chapter this one seems to be culmanating work and the most important part of this is Theorem 5.11
Let F be a field and p(x) an irreducible polynomial in F[x]. Then F[x]/(p(x)) is an extension field of F that contains a root in p(x).
If I understand this correctly this theorem is saying that the congruence class of p(x), in a field F, is the zero class (so [p(x)]=). The example given in the book is which has no roots in . So is an extension of except now there is a root. Specifically .
The chapter talks about how this is enough to establish that exists. While I can see how that is valid I wonder how the extension of a field F is related to the image of a field.