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I have seen, in a few answers on MSE and in uploaded material from some courses, a proof of the primitive element theorem (PET) using Galois theory. It usually goes like this:

Let $F$ be a field and $E$ be a finite separable extension of $F$. By a previous result, it is sufficient (and necessary) that we show that there are finitely many fields intermediate $F$ and $E$. Let $K$ denote the normal closure of $E$ over $F$. By the fundamental theorem of Galois theory, the number of fields intermediate $K$ and $F$ is equal to $|\textrm{Gal}(K:F)|=[K:F]$ which is clearly finite. The result follows immediately.

At the same time, most proofs I've seen of not only the FT of Galois theory, but also those of many preliminary results rely on the PET, so I can't prove the PET using the above route. This got me curious. My question is how far can we really go in Galois theory and field theory in general without direct use of the PET? What about results involving radical extensions?

Since the answer to the 'Galois theory' part of the above question is probably 'reasonably far' (given how common the above proof of the PET is), my next question is how one would accomplish that (looking for references)?

I have already done a few preliminary results, like the fact that imbeddings can be extended to automorphisms of splitting fields or that finite Galois extensions and splitting fields are the same. But I fear I'll hit a roadblock soon.

Lastly, I'd also appreciate rough outlines of the process of these kinds of proofs (or links to such outlines), and I'm interested to know why one would want to proceed without PET in developing Galois theory. Is there any specific motivation for this line of development for the theory?

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    $\begingroup$ My understanding of the historical development is that Emil Artin developed much of modern Galois theory (at least the stuff that undergrads see) with the specific aim of proving the fundamental theorem without the Primitive Element Theorem (which he regarded as unnatural, because it is tantamount to choosing a convenient basis), and he succeeded. I would venture to say that most modern courses in the UK do not use PET (my own undergrad course did not even mention it!). But I will leave it to more confident people to confirm/deny this. $\endgroup$
    – Will R
    Jun 20, 2020 at 17:30
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    $\begingroup$ @JCAA I cannot speak for Artin's state of mind or taste, but at least in his famous booklet on Galois theory, the primitive element theorem is proved much later than the fundamental theorem (theorem 27 vs theorem 16). $\endgroup$ Jun 20, 2020 at 18:23
  • $\begingroup$ @JCAA You know, you should not make hasty assumptions based on your frustration. I did not vote on your answer. $\endgroup$ Jun 20, 2020 at 18:33
  • $\begingroup$ @WillR I see. Yeah, maybe this discrepancy is because I learned Galois theory initially only in fields of characteristic 0 where the PET is much easier to prove and simplifies things immediately. Yes, this makes sense. I have to imagine though that Artin's original work will be difficult to follow for me. Do you know of any accessible references that follow a treatment similar to the one you're familiar with (or provide an outline for a path of proofs)? $\endgroup$ Jun 20, 2020 at 19:16
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    $\begingroup$ @reuns: OP is currently only familiar with proofs of the Fundamental Theorem of Galois theory that use the Primitive Element Theorem. Therefore, OP is confused by the number of times they have seen the Primitive Element Theorem proved using the Fundamental Theorem. Does this clear things up for you? $\endgroup$
    – Will R
    Jun 20, 2020 at 23:19

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Re-posting this comment as an answer, because it's the best answer that's made itself available so-far; but I might come back and edit this if I make time to properly educate myself on this subject!


My understanding of the historical development is that Emil Artin developed much of modern Galois theory (at least the stuff that undergrads see) with the specific aim of proving the fundamental theorem without the Primitive Element Theorem (which he regarded as unnecessary, because it is tantamount to choosing a convenient basis), and he succeeded. I would venture to say that most modern courses in the UK do not use PET (my own undergrad course did not even mention it!).

Source: the comment of JS Milne on this MathOverflow question.

As for mathematical references, the aforementioned Milne has a very nice set of notes on Galois Theory on his website, which I believe takes the approach you are looking for.

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