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I recently found the primitive element theorem a little bit unnatural, and hence I was trying to replace the proofs of some theorems using the primitive element theorem, and here is one.
In the book by Jurgen Neukirch, the theorem that states (1) and (2) are equivalent, and uses the primitive element theorem:
$$\begin{array}{rcl} &L'&\\ &/\backslash\\ L &&K'\\ &\backslash/ \\ &K& \end{array}$$ In this diagram, $L'$ is just the compositum of $L$ and $K'$.

The following are equivalent:

(1) $L|K$ is unramified.

(2) $L'|K'$ is unramified.

And he uses the primitive element to show one implication, while the other is a direct consequence of the first.

When I was trying to deal with this, the concept of value groups came across me, and I am wondering if the unit group of $L|K$ is contained in that of $L'|K'$; if this is the case, then since $L|K$ is unramified, the unit group of $L|K$ is just $\{1\}$, and hence $L'|K'$ is unramified as well.
Nevertheless, this is how far I am now, thanks for any clarification or elaboration.

Edit: Here is a sketch of how the primitive element theorem works:
Fisrtly, since $L|K$ is unramified, the residue class field extension is separable, hence there exists a primitive element of the residue class field extension.

Then we lift it to an element of $L|K$, and use the minimal polynomial of that element to deduce that $[L':K']$ is in fact equal to the degree of the residue class field extension of $L'/K'$, therefore we have proved what was to be proved.

This only serves as a supplement, if anyone needs a full proof, in my opinion, this is contained in every book on the valuation and ramification theory.
Thanks very much.

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    Also, it will be helpful to teach me how to type the notations in the diagram. In any case, thanks very much.2011-02-27
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    @awllower: Please don't use titles as integral information; retype it in the body as a service to the reader. Also, what is $L^{plum}$? What does $plum$ stand for?2011-02-27
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    @allower: This is confusing. What implications? Are you saying he's showing that (1) and (2) are equivalent, and is "plum" meant to be the prime, with $L'=LK'$?2011-02-27
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    I am very sorry, it arose from laziness and ignorance about how to type some specific kind of notations, may I ask if there is a full list of the notations in this program? In any case, thanks very much.2011-02-28
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    @awllower: Thanks for what? You did not answer my question either way; was it or was it not what you meant?2011-02-28
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    @Arturo Magidin: Sorry, may I ask what you want to know?2011-02-28
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    And I have given a description of the notations in the diagram.2011-02-28
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    @awllower: $\overline{F}$ is horrible notation: this is usually reserved for the algebraic closure of a field.2011-02-28
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    @awllower: You did not reply in the comment about what implication, though I see that you have now stated that Neukrich claims the statements "are equal" (which, I'll wager, is *not* what he claims; he must claim that they are *equivalent*; different things).2011-02-28
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    @Arturo Magidin: Sorry but I cannot type other notations, so...2011-02-28
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    @awllower: I don't understand how you can be unable to type other notation. You don't have apostrophes in your keyboard?2011-02-28
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    @Arturo Magidin: Ah, I see it now, it somehow escaped from my sight.2011-02-28

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