I recently came to the topic of field extensions in my abstract algebra course, and there has been a slight issue which has been bothering me that I was hoping I might be able to clear up.

We have defined an extension field for a field F to be a field E such that $F \subseteq E$ and that $F$ is a field under the operations of $E$ restricted to $F$.

Sounds easy enough and I realize that we have been using objects like this for a long time. For example we know that $\mathbb{C}$ is an extension field of $\mathbb{R}$.

Something that has been bothering me a little bit though is that we have started proving theorems where we need to construct extension fields, but these extension fields don't seem to contain the original field $F$ but rather an isomorphic copy of $F$.

For example if our field was $F$, then $F[x]/(p(x))$ is a field if $p(x)$ is irreducible, which contains a subfield isomorphic to $F$. It seems strange that in the theorems (Gallian's Text) that $F[x]/(p(x))$ is considered an extension field for $F$ even though it doesn't really contain $F$ as a subset, but rather another set which is isomorphic.

I don't think I would have normally though this as being much of a problem, but I remember that earlier in the text Gallian seems to mention that even when structures are isomorphic and that they behave essentially the same, that we need to keep in mind that they are not exactly the same.

If this distinction does matter, why not make the definition of an extension field just say that $E$ is an extension field of $F$ if $E$ has a subfield isomorphic to $F$? This would seem to include all cases. Is this largely a historical issue related to how mathematicians thought about isomorphic structures in the past?


Firstly, a short answer. If you are just interested in a field extension of $F$, then you must first realise that you should be quite content with a field extension of any other field $F'$, as long as $F$ and $F'$ are isomorphic and if you know the isomorphism $F\to F'$. This is something we do often in mathematics, yes sometimes without sufficient care, namely ignoring the part that says "if you know the isomorphism...". Often you'd hear people say "we'll just identify these two things since they are isomorphic", though this is not really a healthy thing to do (nor do we actually do that). What we do often is "identify these two things since they are isomorphic and we know precisely which isomorphism we mean for the identification". That is healthy. So, for your field $F$ and the somewhat incorrect claim that $F[x]/(p(X))$ is a field extension of it, what is really going on here is that $F[x]/(p(X))$ is a field extension of an isomorphic copy of $F$, and we know precisely which isomorphism we are talking about, so its ok to identify them. More precisely, we pretend the original $F$ is the isomorphic copy we actually have an extension of.

As long as you are considering just a few objects of study this is usually quite fine. Trouble start when you are considering infinitely many objects. For instance, knowing how to obtain the splitting field extension of a polynomial it is tempting to obtain the algebraic closure of $F$ by 'simply' using a Zorn lemma argument, every time splitting one more polynomial. It is instructive to try and work out the details and see where it fails (lots of difficulties because of those identifications above).

As for you final suggestion to speak of a field extension of $F$ to mean $F'$ contains an isomorphic copy of $F$, you can do that, but you'll have to specify the isomorphism explicitly (since there could be different isomorphic copies, and they can be isomorphic in different ways). But that does not really help, or matter much, since any such superficially broader extension can be replaced, by identifying along the given isomorphism, by the good old notion of extension.


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