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Determination of all Quaternion Octic CM‐Fields with Class Number 2
Author(s) -
Louboutin Stéphane
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/54.2.227
Subject(s) - quaternion , mathematics , algebraic number field , class (philosophy) , field (mathematics) , abelian group , pure mathematics , class number , discrete mathematics , computer science , geometry , quadratic equation , artificial intelligence
It is known that there are only finitely many normal CM‐fields with class number one or with given class number (see [ 9 , Theorem 2; 11 , Theorem 2]) and J. Hoffstein showed that the degree of any normal CM‐field with class number one is less than 436 (see [ 2 , Corollary 2]). Moreover, K. Yamamura has determined all the abelian CM‐fields with class number one: there are 172 non‐isomorphic such number fields. In a recent paper the author and R. Okazaki moved on to the determination of non‐abelian but normal octic CM‐fields with class number one. Noticing that their class numbers are always even, they got rid of quaternion octic CM‐fields, then they focussed on dihedral octic CM‐fields and proved that there are 17 dihedral octic CM‐fields with class number one. The aim of this paper is to get back to the quaternion case: we shall show that there exists exactly one quaternion octic CM‐field with class number 2, namely: Q(√α) with α = −(2 + √2)(3 + √3). Moreover, we shall show that the Hilbert class field of this number field is a normal and non‐abelian CM‐field of degree 16 with class number one.