# Section 5.3: Analysis

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 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)]=[0]). The example given in the book is $p(x)=x^2 + x +1$ which has no roots in $\mathbb{Z}_2$. So $\mathbb{Z}_2 /(x^2+x+1)$ is an extension of $\mathbb{Z}_2$ except now there is a root. Specifically $[p(x)]=[x^2+x+1]=[0]$.

The chapter talks about how this is enough to establish that $\mathbb{C}$ 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.