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

## Theorem 5.11

Let

Fbe a field andp(x)an irreducible polynomial inF[x]. ThenF[x]/(p(x))is an extension field ofFthat contains a root inp(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)]=[0])*. 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.