Say, in a loose sense apparently if $K,L$ are fields and $K \subset L$ I am told that $K$ is a subfield of $L$ and there is a field extension. But the formal definition says,
A field extension is a monomorphism $l:K \rightarrow L$ where $K$ and $L$ are subfields of $\mathbb{C}$.
So, a monomorphism is a map which is one-to-one but not necessarily onto.
So, I should always be able to construct some monomorphism if I am told $L$ is a field extension of $K$.
Specifically, $\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})$. How can there be a monoomrphism $\mathbb{Q} \rightarrow \mathbb{Q}(\sqrt{2})$?
$\mathbb{Q}(\sqrt{2})$ has elements of the form $p+q\sqrt{2}$ where $p,q \in \mathbb{Q}$. So namely, I have to specify some monomorphism $p \rightarrow (p,q)$. I can't think of one explicitly. How can I pinpoint some $p+q \sqrt{2}$ given some $p \in \mathbb{Q}$? This really confuses me, it seems impossible to explicitly write down the monomorphism.
How does this work? Does anyone know an explicit way for example to define a monomorphism here?
"Monomorphism" is not equivalent to "injective" in general, but every homomorphism between fields is automatically injective, which turns out to imply that it's automatically a monomorphism. The (unique!) homomorphism from $\mathbb{Q}$ to $\mathbb{Q}(\sqrt{2})$ sends $a \in \mathbb{Q}$ to $a + 0 \sqrt{2} \in \mathbb{Q}(\sqrt{2})$.