I am reading Abstract Algebra by Hungerford, and I am really confused about how we can extend a ring to a bigger ring. Here's what I got from the book:
$F$ be a field and $p(x)$ be a nonconstant polynomial in $F[x]$. Then, $F[x]/p(x)$ contains a subring $F^*$ that is isomorphic to $F$.
I get that part. But, then it goes:
$F$ be a field and $p(x)$ be a nonconstant polynomial in $F[x]$. Then, $F[x]/p(x)$ is a commutative ring with identiy (OK I see that) containts $F$.
Why contains $F$? We can only know that it would contain something that is isomorphic to $F$, how suddenly that thing becomes $F$? I really need to understand this in order to understand what's going on in the next chapter which discussess about extension field. Any help would be greatly appreciated.
To make myself more clear, let's consider $\mathbb{Z}_3$ which is a field, and consider polynomial $p(x)=x^3+x+2$ which is irreducible in $\mathbb{Z}_3$. Then how can we conclude that $\mathbb{Z}_3\subset\mathbb{Z}_3/p(x)$?
In algebra, we usually study objects (groups, rings, modules, etc.) up to isomorphism. In other words, our theory treats two isomorphic objects as the same, since an isomorphism tells us when two objects look the same as objects. For instance, any ring-related theorem about the integers is true for any ring isomorphic to them. Here we just need to know that there's something ($F^*$) that looks identical (as a field) to $F$ sitting in $F[x]/(p(x))$, so that we can apply our theory about $F$ to that object when we're working in $F[x]/(p(x))$. So in this case, we can just think of $F^*$ as the same object as $F$; we don't really care about the actual symbols that represent $F$.