# Arnie's Monster

## An exceptional totally real cubic field

Predicted by Arnie - discovered by Dan:

In August 1991, I discovered [1] 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 [2] 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 [3]. 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 [3] 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 [1] 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 [4] 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: