# What is the "ray" in ray class group?

I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a person, but I never see it capitalized.

For quick background: the ray class group for a modulus $\mathfrak{m}$ of a number field $K$ is the group of fractional ideals of $K$ prime to $\mathfrak{m}$ modulo the principal ideals generated by elements of $K$ congruent to $1$ modulo $\mathfrak{m}$, where "congruent to $1$" for a real place means "positive."

• It is certainly not a person because it is called rayon in French. Jul 9, 2012 at 3:09
• Well, people exist who think that Killing vector fields is some kind of action-movie motivated term... :) Jul 9, 2012 at 3:20
• ...and I have seen Lie groups translated as groupes de mensonge. Jul 9, 2012 at 7:09
• In my first microsecond of reading this question, I thought it was related to the eternal question of citizens of New York City: who was Ray of Ray's Pizza? Jul 9, 2012 at 21:02

There are many introductions to number theory, but few are as original (my term for what others would call weird) as Fueter's "Synthetische Zahlentheorie" published in 1925. It starts with elementary number theory, and discusses the arithmetic of cyclotomic fields up to the Dedekind zeta function and applications to quadratic and cubic reciprocity.

In § 5, Fueter defines rays in the field of rational numbers:

Definition. If a set of numbers has the property that it contains the product and the quotient of any two of its elements, then this set is called a ray.

Actually Fueter does not use the word "set" [Menge] but rather talks about a "domain of numbers" [Bereich von Zahlen]. At the end of this paragraph he makes the following historical remarks:

The necessity of considering sets with the property of rays was first realized by Weber. He called these sets "number groups". Independently, these groups were introduced by R. Fueter (Der Klassenkörper der quadratischen Körper und die komplexe Multiplikation, Diss. Univ. Göttingen, 1903; see also Crelle 130 (1905), p. 208), who called them rays. We keep this name here since it is similar in nature to the geometric names of fields [in German: Körper, i.e., solid] and ring, and since the word "group" does not imply commutativity, which is always satisfied by rays.

I would have been surprised had there been connections with infinite primes; back in 1903, Hilbert had already defined infinite primes, but their prominent role in class field theory only became apparent through the work of Furtwängler and Takagi, which took place after the term "ray" had been coined.

• It is a real pleasure to have you among the regular MO users, Franz! Jul 10, 2012 at 0:03

As noted in the other answers "ray" comes from the German word "Strahl". In his Zahlbericht p.5 (can be found here: http://www.digizeitschriften.de/index.php?id=resolveppn&PPN=GDZPPN002127768) Hasse wrote:

Zweitens kann man gewissermaßen das Fundament für diese Bildungen tiefer legen, indem man anstelle von $H_0$ gewisse Untergruppen von $H_0$, die sogenannten Strahlen, zugrunde legt [...] Ist $m$ irgendein ganzes Ideal aus $k$, so verstehen wir unter dem Strahl mod. $\mathbf{m}$ die folgendermaßen entstehende Idealgruppe: Man lasse $\alpha$ alle den Bedingungen $$\alpha \equiv 1 \text{ mod.}\; m;\;\; \alpha \gg 0\;\;\; \text{(total-positiv)}$$ genügenden Zahlen aus $k$ durchlaufen, und bilde jedesmal das zugehörige Hauptideal $(\alpha)$.

Here $H_0$ is the subgroup of all principal ideals and Hasse defines the ray mod $m$ as the subgroup of $H_0$ generated by all principal ideals $(\alpha)$ where $\alpha \in k$ satisfies the quoted condition.

As explained by Filippo, the set of all reals $\alpha > 0$ is called a "Strahl" in geometry. Therefore it seems pretty clear to me that the name "Strahl" in the quoted text was used because the $\alpha$'s in the definition of the rays lie in a "Strahl" in geometric sense.

• Nice answer. Do you know if "Teil II" of the Zahlenbericht is available online? I could only find "Teil I" and "Teil Ia". Jul 9, 2012 at 15:24
• I didn't find it online. The point is that part I and part Ia is published in the DMV journal (whose issues are online) while part II seems to be printed as a whole book in its own (the DMV Ergänzungsband 6 from 1930, in 1965 also published as 2nd edition by Physica-Verlag). Jul 9, 2012 at 17:47
• Ulf Rehmann (Bielefeld) had promised some time ago that he and Keith Dennis (Cornell) will put Teil II online but it doesn't seem to have happened yet. Jul 10, 2012 at 5:07

A "ray class group" is constructed from a "ray" $K_{m,1}$, so the question is why $K_{m,1}$ is called a ray. I don't have direct evidence for this, but it seems pretty clear to me that the reason for this terminology is that the condition coming from a real primes in m is that $x \in K_{m,1}$ lies in the ray (in the high school geometry sense) of positive real numbers (under the appropriate real embedding).

The word ray comes from the German Strahl, as in Strahlklassengruppe which Hasse uses but which goes back to Fueter, as Franz points out in a comment below.

• That's helpful, but we still have the question of why the term ray (or Strahl) is used. Noah's idea seems the most convincing, but I'm still looking for evidence. Jul 9, 2012 at 6:01
• I always thought that the expression Strahl was coined by Fueter in his early articles on complex multiplication. If I recall it correctly, Weber talked only about class groups of orders and ring class groups. Jul 9, 2012 at 9:38
• If you thought so Franz, it must be correct. I mentioned Weber only because the concept goes back to him. Jul 9, 2012 at 9:57
• Dear Chandan, You have a typo: "Feuter" --> "Fueter". Best wishes, Matthew Jul 9, 2012 at 15:08

In German Strahl means ray but is mathematically used to mean a half-line, infinite in only one direction. I guess that the use of Strahlklassengruppe refers to the fact that the ray class field modulo $\mathfrak{m}$ "begins" the line of tower extensions unramified outside of $\mathfrak{m}$.

• @Filippo: What is Strahlklassengruppe in Italian ? Jul 9, 2012 at 4:01
• @Dalawat: Hi Dalawat! We simply say "gruppo delle classi modulo $\mathfrak{m}$", thus omitting completely anything involving Strahl, ray, rayon, etc... Jul 9, 2012 at 4:22