# Hilbert’s Theorem 90

That’s right. Today I’m going to talk about where my blogging name (and basically my online name everywhere) came from.

If you go up to a mathematician and ask if they know who David Hilbert is, chances are they will say, “Yes! He is one of the most famous mathematicians of all time.” If they are problem oriented, then they will probably go off on his 23 problems that were presented at the International Congress of Mathematics in 1900. They may even say that this is what the current “Millenium problems” are based on.

On the other hand, David Hilbert is quite well-known to philosophers of math and science as the great formalist. He wanted to completely axiomatize mathematics. He was concerned with constructive proofs from these foundations. People on this side of the fence probably are familiar with his quote, “Wir mussen wissen. Wir werden wissen.” (or “We must know. We will know.”)

I want to present some history that is lesser known. There are many theorems associated with Hilbert. People familiar with algebraic number theory are used to seeing things like “HIlbert’s Theorem 90” or “Hilbert’s Theorem 92” (often also called Hilbert Satz 90 …) etc. But what are these referring to? It can’t possibly be his 90th theorem.

These theorems are in reference to Hilbert’s book Zahlbericht. He wrote the text in 1897, and it was the number theory text that many famous mathematicians such as Artin, Hasse, Hecke, and Weyl used (it was basically the only modern treatment of algebraic number theory available). One should note that this is remarkable considering “bericht” means “report,” and Hilbert literally wrote this text as a report of the state of algebraic number theory for the German Mathematical Society.

When Hilbert was comissioned to write the report, he was to work with Minkowski to write the state of all of number theory. Minkowski’s half was never written.

It is also rather interesting to note that although Hilbert had done work in algebraic number theory, this was a report on all of number theory, so the intent was not original work. In fact, the work that Hilbert is most known for in the field had not even been done yet. The report was designed to give direction to the field. It is because of this that it received much criticism most notably from Kummer who blamed Hilbert for the delays in some of his publications. This is probably a valid argument considering Hilbert often criticized Kummer for using complicated computations and even said that he avoided a lot of his work in the report. Hilbert even went ahead and replaced many of Kummer’s proofs with his own (e.g. satz 166-171 are Kummer’s theorems but Hilbert’s proofs).

One interesting historical point is how our notation and terminology has evolved over the past 100 years. For instance, many simple ideas from algebra had not been formalized yet such as a quotient group. At one point Hilbert writes, “the members of G are obtained precisely once when we multiply the members of H by $1, g, g^2, \ldots, g^{n-1}$ where g is a suitably chosen member of G.” In our current terminology we would just say, “$G/H$ is a finite cyclic group.”

On the other hand we now get to Hilbert’s Theorems 89-94, which are considered to have influenced the subject in a very positive way. The 90th theorem (my name!) is arguably the most famous. This actually is one of Kummer’s. I won’t go into the original, but in modern terminology it is: If $K$ is a finite Galois extension of $F$ with $G=Gal(K/F)$ cyclic of order n (with generator $\sigma$), then if $a\in K$, then $N_{K/F}(a)=1$ if and only if $a=b/(\sigma(b))$ for some $b\in K$. A generalization in terms of cohomology was found by Emmy Noether.

So next time you run into a theorem labelled “Hilbert Theorem ___” or “Hilbert Satz ____” you know where it came from. You also know that it is very likely that it is not Hilbert’s theorem at all, but was just compiled by him for a report.

Citations: I must admit that little is published on this subject and everything I wrote here I got from the introduction to the english edition of the Zahlbericht by Lemmermeyer and Schappacher. The exact statement of the 90th satz was taken from Larry Grove’s Algebra.