Talk:Primitive element theorem

Before Artin

With all due respect to Emil Artin, there were treatments of field theory before him, and after him. I have therefore added a reference to clarify the history. I have also rearranged the article and added some considerations about being constructive. Charles Matthews (talk) 11:37, 17 April 2010 (UTC)[reply]

You are right. Actually, though the statements were stated and proved in Emil Artin's book "Galois Theory" from 1942, what is here called Artin's theorem should be rather called Steinitz' theorem, since Ernst Steinitz in his seminal paper "Algebraische Theorie der Körper" (Algebraic theory of fields), published in Crelle's journal (J. Reine Angew. Math.) in 1910 (!), has stated and proven this theorem. (He calls it "theorem of the intermediate fields", and that, what is called Corollary here, "theorem of the primitive elements".) I do not know for sure, but very likely Steinitz was the first who stated and proved this (at least in this gemerality and clarity). On the other hand, existence of primitive elements (in a more special situation) traces back to Lagrange and Galois. So, this should be fixed. Additionally, I think it should be eplained, in which way the corollary is really such. What is ("allowed" to be) used in the proof? DrKssn (talk) 22:24, 2 April 2021 (UTC)[reply]

Transcendental extensions

I do not understand the phrase "in this situation". A transcedental extension with primitive element x say or a transcendental number does not allow every element to be expressed as a poynomial in the primitive element as the article seems to say. Bukovets (talk) 11:46, 19 February 2011 (UTC)[reply]

Merge with (or from) simple extension

It seems to me that the two articles, this one and "simple extension" cover the same materials to very large extent; thus, some form of merger would make sense. I'm not sure about the direction of merger. While "simple extension" is a broader concept, the title of this article is very common. -- Taku (talk) 19:30, 18 April 2013 (UTC)[reply]

Correction needed

As "Bukovets" pointed out above, there seems to be a mistake in the current article (under "terminology"). In the simple extension F(x) it is not necessarily the case that every element can be presented as a polynomial in x with coefficients from F (i.e. when x is transcendental over F). 217.255.103.87 (talk) 20:45, 3 August 2013 (UTC)[reply]

Correction provided

Addressed "Bukovets" 's correct concern by removing the example of a transcendental/indeterminate extension. Ncsinger (talk) 16:04, 25 May 2016 (UTC)[reply]

Separation of the article?

Hello there,

I have come to the belief that this article should be separated, possibly using a disambiguation page.

This is because two theorems are being treated:

The one of Steinitz, which states that the extension is simple iff there are finitely many intermediate fields,
and the one of Galois, which states that any separable, finite extension is simple.

I'm tending towards moving Steinitz' theorem to a separate article, even though the lemma already exists. In this case, one would have to introduce a disambiguation page for this lemma. --Mathmensch (talk) 20:17, 3 December 2023 (UTC)[reply]

The main problem here is that the sources differ on which of the two is called the "primitive element theorem". I believe to have discerned a slight tendency towards Galois' result. --Mathmensch (talk) 20:36, 3 December 2023 (UTC)[reply]