The determination of the imaginary abelian number fields with class number one
نویسندگان
چکیده
منابع مشابه
The Determination of the Imaginary Abelian Number Fields with Class Number One
In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 • 163 , which is the conductor of the biquadratic number field Q(\/-67, v'—163).
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Let N be an imaginary abelian number field. We know that hN , the relative class number of N , goes to infinity as fN , the conductor of N , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. I...
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Mathematics Subject Classi cation: Primary, 11R20, 11R29, 11Y40; Secondary, 11M20, 11R42.
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because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward through the main storyline without b...
متن کاملThe Number of Imaginary Quadratic Fields with a given Class Number
Gauss asked for a list of imaginary quadratic fields with class number one. This problem inspired a great deal of work; some of the prominent milestones being the work of Heilbronn showing that there are only finitely many fields with a given class number, the work of Landau and Siegel providing good (but ineffective) lower bounds for the class number, the work of Heegner and Stark showing that...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1994
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-1994-1218347-3