On the class numbers of cyclotomic fields

Horie, Kuniaki

doi:10.1007/bf01172792

Cited by 7 publications

(3 citation statements)

References 9 publications

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“…Now, suppose that the ideal class group of K is of exponent < 2. Using A*(K) < 2JVo'(->") and (6), we get (8) (_ML_Y <9 37Vl0g(p2/,) Hence, we get /(0) > /(l) > /(2). On the other hand, since we have TV > 4, log(p2fr) > log(52) and x i-> (x2 -l)log(52) -log(4x) is a positive (and increasing) function on [(5/4), +00), we get f(r+l) > f(r) for r > 2.…”

Section: Lemma (I) (See [13 Lemma 1]) Let Xp Be a Primitive Dirichlmentioning

confidence: 86%

Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2

Louboutin¹

1995

Mathematics of Computation

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We determine all nonquadratic imaginary cyclic number fields K of 2-power degrees with ideal class groups of exponents < 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some (Z/2Z)m , m > 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).

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Section: Lemma (I) (See [13 Lemma 1]) Let Xp Be a Primitive Dirichlmentioning

confidence: 86%

Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2

Louboutin¹

1995

Mathematics of Computation

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“…Horie [1] proved that there are only finitely many pairs On the other hand, it is known that h + p = 4 under GRH by van der Linden [7] and h − p = 4 · N for some odd integer N . Therefore, it follows that θ r,H kills the q-part Cl K (q) for any odd prime number q (under GRH).…”

Section: Lemma 2 Let Q Be An Odd Prime Number Let E/f Be a Cyclic Ementioning

confidence: 97%

Hilbert–Speiser number fields for a prime p inside the p-cyclotomic field

Ichimura

2008

Journal of Number Theory

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“…Now, suppose that the ideal class group of K is of exponent < 2. Using A*(K) < 2JVo'(->") and (6), we get (8) (_ML_Y <9 37Vl0g(p2/,)…”

Section: The Case P /mentioning

confidence: 99%

Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤2

Louboutin¹

1995

Math. Comp.

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Abstract.We determine all nonquadratic imaginary cyclic number fields K of 2-power degrees with ideal class groups of exponents < 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some (Z/2Z)m , m > 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).

show abstract

On the class numbers of cyclotomic fields

Cited by 7 publications

References 9 publications

Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2

Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2

Hilbert–Speiser number fields for a prime p inside the p-cyclotomic field

Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤2

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