Ray in Ray Class Group – Number Theory and History

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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."

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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.

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Understanding the Class Number in Number Theory

This is from Buell, Binary Quadratic Forms. From page 84, the class number for a negative discriminant $\Delta$ is about $$\frac{\sqrt{|\Delta|}}{\pi},$$ which comes from an $L$-function calculation on page 83.

Let's see, on page 101, he points out that for negative field discriminants, class group and narrow class group are identical. Then on page 103, the group of classes of binary quadratic forms is isomorphic to the narrow class group. So that works out.

I don't know about surjectivity of class numbers. I imagine so. See OEIS

I wrote a little program up to 1000, here it is up to 111. The first number that achieves a given class number tends to be squarefree, an exception being h=104. EDIT: I've run this up to 4000 so far. To get a class number $h,$ there was always some $k$ with $k < 4 h^2.$ The largest $h$ where $h^2$ did not suffice was $h=677,$ with smallest $k = 601247.$ For $678 \leq h \leq 4000,$ there was always some $k \leq 0.751517... h^2,$ equality at $h=857, k=551951.$

    1       3 = 3
    2      15 = 3 * 5
    3      23 = 23
    4      39 = 3 * 13
    5      47 = 47
    6      87 = 3 * 29
    7      71 = 71
    8      95 = 5 * 19
    9     199 = 199
   10     119 = 7 * 17
   11     167 = 167
   12     231 = 3 * 7 * 11
   13     191 = 191
   14     215 = 5 * 43
   15     239 = 239
   16     399 = 3 * 7 * 19
   17     383 = 383
   18     335 = 5 * 67
   19     311 = 311
   20     455 = 5 * 7 * 13
   21     431 = 431
   22     591 = 3 * 197
   23     647 = 647
   24     695 = 5 * 139
   25     479 = 479
   26     551 = 19 * 29
   27     983 = 983
   28     831 = 3 * 277
   29     887 = 887
   30     671 = 11 * 61
   31     719 = 719
   32     791 = 7 * 113
   33     839 = 839
   34    1079 = 13 * 83
   35    1031 = 1031
   36     959 = 7 * 137
   37    1487 = 1487
   38    1199 = 11 * 109
   39    1439 = 1439
   40    1271 = 31 * 41
   41    1151 = 1151
   42    1959 = 3 * 653
   43    1847 = 1847
   44    1391 = 13 * 107
   45    1319 = 1319
   46    2615 = 5 * 523
   47    3023 = 3023
   48    1751 = 17 * 103
   49    1511 = 1511
   50    1799 = 7 * 257
   51    1559 = 1559
   52    1679 = 23 * 73
   53    2711 = 2711
   54    2759 = 31 * 89
   55    4463 = 4463
   56    1991 = 11 * 181
   57    2591 = 2591
   58    2231 = 23 * 97
   59    2399 = 2399
   60    2159 = 17 * 127
   61    3863 = 3863
   62    2471 = 7 * 353
   63    2351 = 2351
   64    2519 = 11 * 229
   65    3527 = 3527
   66    3431 = 47 * 73
   67    3719 = 3719
   68    2831 = 19 * 149
   69    3119 = 3119
   70    3239 = 41 * 79
   71    5471 = 5471
   72    3311 = 7 * 11 * 43
   73    2999 = 2999
   74    4151 = 7 * 593
   75    4703 = 4703
   76    3071 = 37 * 83
   77    6263 = 6263
   78    5111 = 19 * 269
   79    4391 = 4391
   80    5183 = 71 * 73
   81    3671 = 3671
   82    3839 = 11 * 349
   83    3911 = 3911
   84    4031 = 29 * 139
   85    4079 = 4079
   86    6767 = 67 * 101
   87    5279 = 5279
   88    4199 = 13 * 17 * 19
   89    6311 = 6311
   90    5951 = 11 * 541
   91    4679 = 4679
   92    4991 = 7 * 23 * 31
   93    5351 = 5351
   94    7367 = 53 * 139
   95    6959 = 6959
   96    6071 = 13 * 467
   97    5519 = 5519
   98    6191 = 41 * 151
   99    5591 = 5591
  100    7991 = 61 * 131
  101    5879 = 5879
  102    9383 = 11 * 853
  103   13799 = 13799
  104    9359 = 7^2 * 191
  105    6719 = 6719
  106    7631 = 13 * 587
  107    8231 = 8231
  108    5759 = 13 * 443
  109    5711 = 5711
  110    7751 = 23 * 337
  111   15359 = 15359

[Math] Class numbers of orders

The ideal class semigroups mentioned in the question got studied in this setting (orders of number fields) and in other and more general ones by various authors in recent years.

A starting references is:

Zanardo, P.; Zannier, U. The class semigroup of orders in number fields. Math. Proc. Cambridge Philos. Soc. 115 (1994), 379–391.

They show that this semigroup is a Clifford semigroup for orders of quadratic fields, but for degree greater 2, there always is some order such that the respective class semigroup is not a Clifford semigroup.

There is also an older paper on this theme (that it appears was overlooked in recent literature for some time but reapperas in still more recent one)

Dade, E. C.; Taussky, O.; Zassenhaus, H. On the theory of orders, in paricular on the semigroup of ideal classes and genera of an order in an algebraic number field. Math. Ann. 148 (1962) 31–64.

Starting from these two papers and looking for papers that quote them in MathSciNet for example one will find several more recent contributions. Investigating these semiclass groups; but it seems (but I donot have a good overview) the emphasis is more to generalize to more general structures (say, other domains than just orders) than more details in the number-theoretic setting.

Yet, in particular the early papers (I do not know for the recent ones) contain also some explicit information on these semigroups, but as commented earlier I would not know of in some sense simple descriptions (but again my knowledge is superficial here).

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Understanding the Class Number in Number Theory

[Math] Class numbers of orders

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