# Arnie's Monster

## An exceptional totally real cubic field

Predicted by Arnie - discovered by Dan:

In August 1991, I discovered  that d = 146853 = 632 x 37 is the smallest (and up to d = 200000 unique) totally real cubic discriminant which verifies a Conjecture of Arnold Scholz  that the class number relation h(N) = h(k)h(L)2 / 9 should also be possible for ring class fields N|k with conductor f<>1. Up to my discovery, the validity of this class number relation was warranted only for certain absolute (Hilbert) class fields N | k with conductor f = 1. Here, N denotes the sextic normal field of the totally real cubic field L with discriminant d and k the real quadratic subfield of N.

Relative Principal Factorization instead of Capitulation:

Indeed, the conductor of the cubic relative extension N | k associated with the totally real cubic field L with discriminant d = 146853 is f = 63. It has the two prime divisors 3 and 7 which both split in the real quadratic field k with discriminant 37. Since the class number of k is h(k) = 1, we cannot have a principalization (capitulation of ideal classes) in N | k. Instead we observe a 2-dimensional relative principal factorization generated by the prime ideals lying over 3 and 7 in the sextic normal field N. A field L with this property was said to be of principal factorization type ALPHA 3 in . Arnie's Monster is the first and, up to now, unique known case of type ALPHA 3.

References:

A description of all possible principal factorization types that have been defined in  you will find on the page Principal Factorization Types of Cubic Fields.

The complete statistics of the population of these principal factorization types in the range of the computations in  you get by linking to the table Distribution of Totally Real Cubic Fields.

The Ideal Class Group of Arnie's Monster:

Already in 1985, our attention has been drawn to the totally real cubic field L with discriminant d = 146853, when ENNOLA and TURUNEN  found that it is one of the few fields up to d = 500000 with a non-cyclic 3-class group C(3) x C(3).
So there is possibly a connection between the rare principal factorization type ALPHA 3 and a non-cyclic 3-class group.

Bibliography: