This is the excerpt from Lang's Algebra book and some moments seem to very confusing.
1) If $E,F$ are fields such that $E,F\subset L$ then he defines $EF$ as the smallest subfield of $L$ which contains both $E$ and $F$. Generally speaking, this $EF$ should be subset of $L$, right?
2) Then he shows that $EF$ is the field of fractions of $E[F]$. But in this case $EF$ is no more the subset of $L$ since in that case elements of $EF$ are equivalence classes. That's really confuses me and my understanding of the context.
3) How to show correctly and rigorously that the equality which I've underlined is true. Since $\dfrac{a_1b_1+\dots+a_nb_n}{a'_1b'_1+\dots+a'_mb'_m}$ is the equivalence class.
Would be very grateful for answering my questions! Please help.
1) Yes, $EF$ will be a subset of $L$. After all, $L$ is a field containing $E$ and $F$, so the smallest such field will be a subset.
2) OK, the field of fractions as constructed is no longer in $L$, depending on how we interpret that ring. But that's what homomorphisms are for - like that embedding $\sigma$ introduced in the next lemma. We have natural maps from $E$ and $F$ into $L$, which can be naturally extended to map the ring $E[F]$ and its field of quotients into $L$.
You object based on the notion that the elements of $EF$ are actually equivalence classes - OK, that just means we have to show that the homomorphism maps each equivalence class to a single element. If $\sigma$ maps $E(F)$ to $L$, then we extend it by defining $\sigma\left(\frac{u}{v}\right) =\frac{\sigma(u)}{\sigma(v)}$. If $\frac uv$ and $\frac{u'}{v'}$ are equivalent, that's $uv'=u'v$, and $\sigma\left(\frac{u}{v}\right)-\sigma\left(\frac{u'}{v'}\right) = \frac{\sigma(u)}{\sigma(v)}-\frac{\sigma(u')}{\sigma(v')}=\frac{\sigma(u)\sigma(v') - \sigma(u')\sigma(v)}{\sigma(v)\sigma(v')} = 0$. Done.
3) The underlined equality is simply how (field) homomorphisms work; they distribute over all arithmetic operations, so applying them to the whole thing is the same as applying them to every little piece. (OK, putting $\sigma$ in the exponents is weird notation - I'm not sure what's up with that.)