Hilbert's basis theorem original formulation.

Question

Hilbert's basis theorem (1888) is usually stated as: "If R is a Noetherian ring, then R[X] is a Noetherian ring."

This could not be the original formulation of the theorem since Noetherian rings were named after Emmy Noether, who lived from 1882 to 1935.

Do you know the original formulation of the theorem? Or, even better, can you point me out a reference to find it?

Answer 1

We go to the wiki article and find:

Hilbert (1890) proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.

And look, the 1890 is a link to the publication information

Hilbert, David. "Über die Theorie der algebraischen Formen." Mathematische annalen 36.4 (1890): 473-534.

A search for that gets you a link.

This will probably confirm what the wiki article says. There are many places on the internet to perform a machine translation. I bet there is already a translated version out there.

So there you see, it's almost as if the information wanted to be found. You just have to look for it.

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Give a man a fish and they eat for a day, teach a man to fish and you get a downvote. Oh well! Enough time has passed I suppose there's no harm in elaborating. I am interested in the topic myself. All translations courtesy of google translate, and I assume $k$ is a field.

Odds are the relevant theorem is Theorem 1, the one preceded by the phrase

In order to decide this question, we first prove the following theorem, which is fundamental for our further investigations:

The theorem then is:

If any non-terminating series of forms of the $n$ variables $x_1,x_2,\ldots,x_n$ is presented, say $F_1,F_2, F_3\ldots$, there is always a number $m$ such that every form of that series can be brought into the form $$ F=A_1F_1+A_2F_2+\cdots+A_mF_m $$ where $A_1, A_2, \ldots A_m$ suitable forms of the same $n$ variables are.

Given the context hints from the wiki article, I'd say this can be refined to

If any non-terminating series of polynomials in $k[x_1,x_2,\ldots,x_n]$ is given, say $F_1,F_2, F_3\ldots$, there is always a number $m$ such that every polynomial of that series can be brought into the form $$ F=A_1F_1+A_2F_2+\cdots+A_mF_m $$ where $A_1, A_2, \ldots A_m$ suitable elements of $k[x_1,x_2,\ldots, x_n]$.

(In fact, the current wikipedia article on Noetherian rings explicitly confirms that this is Hilbert's original version.)

I think the most useful rephrasing of this is

Every countable increasing chain of finitely generated ideals of $k[x_1,x_2,\ldots,x_n]$ stabilizes.

This is a well-known equivalent condition to the Noetherian condition, since it says all ideals are finitely generated.