The following is a paragraph of a text I'm not understanding
"Definition: An isomorphism between two field extensions $i:K→K'$, $j:L→L'$ is a pair $(\alpha, \beta)$ of field isomorphisms $\alpha:K→L$, $\beta:K'→K$, such that
$$j\big(\alpha(k)\big)=\beta\big(i(k)\big)$$
(...)
Various identifications may be made. If we identify $K$ and $i(K)$, and $L$ and $j(L)$, then $i$ and $j$ are inclusions, and the commutativity condition now becomes
$$\beta|_K=α$$
where $\beta|_K$ denotes the restriction of $\beta$ on $K$."
There are a few things I don't get about the latter text.
$Q1$: What is the meaning of 'identifying' a Field? If it's not too much to ask, could someone provide some example, they usually help a lot.
$Q2$: to say that $i$ and $j$ are inclusion is simply to say that (I guess that under some circumstances) $i(k) = k$, for $k∈K$ right?
$Q3$: to say that $\beta|_K=\alpha$ simply means that (I guess, again, that under some circumstances) $\beta(k) = \alpha(k)$, for $k∈K$, right?
I would really appreciate any help/thoughts
1) Making helpful (but technically incorrect) identifications is very common and useful in algebra. For example, $\mathbb{Z}[x]$, the ring of polynomials in $x$ with integer coefficients doesn't technically contain $\mathbb{Z}$ but we make the identification of $\mathbb{Z}$ with the constant polynomials in $\mathbb{Z}[x]$. A field identification is an analogous concept.
2). Yes
3). Yes