In my galois theory class, on a review sheet, my professor wrote: Determine with proof whether the following rings are isomorphic to subrings of $\mathbb{R}$.
- $\frac{\mathbb{Q}[x]}{<x^2-2x-2>}$
- $\frac{\mathbb{Q}[x]}{<x^2-2x+1>}$
- $\frac{\mathbb{Q}[x]}{<x^2-2x+2>}$
For (1), I understand that the ideal is irreducible, and hence we have a field. From this using that that any homomorphism $\phi:K\rightarrow \mathbb{R}$ where $K$ is a field is an injection from $K$ to $\phi(K)$. My professor used that a homomorphism is uniquely determined by the image of $x$. If we let $\phi(x)=1+\sqrt{3}$, we define a homomorphism. Does $K$ need to be a field for this to be true? How can I be sure that his is a homomorphism as well? (my professor claimed that it is a homomorphism without checking, so I assume there is a reason.) Additionally, what sort of tools will I need for the remaining problems. I am sort of behind and I would really appreciate it if a sort of comprehensive set of tools for these problems could be given.
Let $K \subset F$ be fields and $f \in K[x]$ irreducible. Then I claim that $K[x]/(f)$ is isomorphic to a subfield of $F$ if and only if $f$ has a root in $F$.
On the one hand, $K[x]/(f)$ clearly contains a root of $f$. So if it is a subfield of $F$, $F$ clearly also contains a root of $f$.
On the other hand if $a \in F$ with $f(a)=0$, we have a well-defined injection $$K[x]/(f) \hookrightarrow F, x \mapsto a.$$