Asymptotic Results for Class Number Divisibility in Cyclotomic Fields

1983 ◽  
Vol 26 (4) ◽  
pp. 464-472 ◽  
Author(s):  
Frank Gerth
Keyword(s):  
Lower Bounds ◽  
Class Number ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Cyclotomic Fields ◽  
Asymptotic Results ◽  
Root Of Unity ◽  
Almost All

AbstractLet n ≥3 and m≥3 be integers. Let Kn be the cyclotomic field obtained by adjoining a primitive nth root of unity to the field of rational numbers. Let denote the maximal real subfield of Kn. Let hn (resp., ) denote the class number of Kn (resp., ). For fixed m we show that m divides hn and hn for asymptotically almost all n. Also for those Kn and with a given number of ramified primes, we obtain lower bounds for certain types of densities for m dividing hn and .

On the Class-Number of the Maximal Real Subfield of a Cyclotomic Field

1981 ◽  
Vol 33 (1) ◽  
pp. 55-58 ◽  
Author(s):  
Hiroshi Takeuchi
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Cyclotomic Field ◽  
Rational Numbers ◽  
Real Quadratic Field ◽  
The Real ◽  
Root Of Unity ◽  
Image Position ◽  
Real Quadratic

Let p be an integer and let H(p) be the class-number of the fieldwhere ζp is a primitive p-th root of unity and Q is the field of rational numbers. It has been proved in [1] that if p = (2qn)2 + 1 is a prime, where q is a prime and n > 1 an integer, then H(p) > 1. Later, S. D. Lang [2] proved the same result for the prime number p = ((2n + 1)q)2 + 4, where q is an odd prime and n ≧ 1 an integer. Both results have been obtained in the case p ≡ 1 (mod 4).In this paper we shall prove the similar results for a certain prime number p ≡ 3 (mod 4).We designate by h(p) the class-number of the real quadratic field

The Class Number of the Cyclotomic Field

1951 ◽  
Vol 3 ◽  
pp. 486-494 ◽  
Author(s):  
N. C. Ankeny ◽  
S. Chowla
Keyword(s):  
Class Number ◽  
Cyclotomic Field ◽  
Root Of Unity

Let g denote an odd prime, and h = h(g) the class number of the cyclotomic field R(), where is a primitive gth root of unity. It is known that we can write

scholarly journals Power integral bases for real cyclotomic fields

2005 ◽  
Vol 71 (1) ◽  
pp. 167-173 ◽  
Author(s):  
Laurel Miller-Sims ◽  
Leanne Robertson
Keyword(s):  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Integral Basis ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Root Of Unity ◽  
Integral Bases

We consider the problem of determining all power integral bases for the maximal real subfield Q (ζ + ζ−1) of the p-th cyclotomic field Q (ζ), where p ≥ 5 is prime and ζ is a primitive p-th root of unity. The ring of integers is Z[ζ+ζ−1] so a power integral basis always exists, and there are further non-obvious generators for the ring. Specifically, we prove that if or one of the Galois conjugates of these five algebraic integers. Up to integer translation and multiplication by −1, there are no additional generators for p ≤ 11, and it is plausible that there are no additional generators for p > 13 as well. For p = 13 there is an additional generator, but we show that it does not generalise to an additional generator for 13 < p < 1000.

scholarly journals Primes in Short Segments of Arithmetic Progressions

1998 ◽  
Vol 50 (3) ◽  
pp. 563-580 ◽  
Author(s):  
D. A. Goldston ◽  
C. Y. Yildirim
Keyword(s):  
Asymptotic Formula ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Riemann Hypothesis ◽  
Total Variance ◽  
Correct Order ◽  
Asymptotic Results ◽  
Extended Range ◽  
Order Of Magnitude ◽  
Almost All

AbstractConsider the variance for the number of primes that are both in the interval [y,y + h] for y ∈ [x,2x] and in an arithmetic progression of modulus q. We study the total variance obtained by adding these variances over all the reduced residue classes modulo q. Assuming a strong form of the twin prime conjecture and the Riemann Hypothesis one can obtain an asymptotic formula for the total variance in the range when 1 ≤ h/q ≤ x1/2-∈ , for any ∈ > 0. We show that one can still obtain some weaker asymptotic results assuming the Generalized Riemann Hypothesis (GRH) in place of the twin prime conjecture. In their simplest form, our results are that on GRH the same asymptotic formula obtained with the twin prime conjecture is true for “almost all” q in the range 1 ≤ h/q ≤ x1/4-∈, that on averaging over q one obtains an asymptotic formula in the extended range 1 ≤ h/q ≤ x1/2-∈, and that there are lower bounds with the correct order of magnitude for all q in the range 1 ≤ h/q ≤ x1/3-∈.

A note on the integral trace form in cyclotomic fields

2015 ◽  
Vol 14 (04) ◽  
pp. 1550045 ◽  
Author(s):  
J. Carmelo Interlando ◽  
Trajano Pires da Nóbrega Neto ◽  
Tatiana Miguel Rodrigues ◽  
José Othon Dantas Lopes
Keyword(s):  
Nonzero Element ◽  
Cyclotomic Field ◽  
Explicit Description ◽  
Complex Conjugate ◽  
Cyclotomic Fields ◽  
Trace Form ◽  
Root Of Unity

Let m ≥ 3 be an integer, ζm ∈ ℂ a primitive mth root of unity, and Km the cyclotomic field ℚ(ζm). An explicit description of the integral trace form [Formula: see text] where [Formula: see text] is the complex conjugate of x is presented. In the case where m is prime, a procedure for finding the minimum of the form subject to x being a nonzero element of a certain ℤ-module in ℤ[ζm] is presented.

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

2014 ◽  
Vol 10 (02) ◽  
pp. 283-296
Author(s):  
HUMIO ICHIMURA ◽  
SHOICHI NAKAJIMA ◽  
HIROKI SUMIDA-TAKAHASHI
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Relative Class ◽  
Quadratic Twists ◽  
Large N ◽  
Relative Class Number

Let p be an odd prime number, Kn = Q(ζpn+1) the pn+1th cyclotomic field and [Formula: see text] the relative class number of Kn. Fixing an integer d ∈ Z with [Formula: see text], we denote by Ln the imaginary quadratic subextension of the imaginary (2, 2)-extension [Formula: see text] with Ln ≠ Kn. When d < 0, we have [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the minus parts of the 2-adic Iwasawa lambda invariants of Kn and Ln, respectively. By a theorem of Friedman, these invariants are stable for sufficiently large n. First, under the assumption that [Formula: see text] is odd for all n ≥ 1, we give a quite explicit version of this result. Second, we show that the assumption is satisfied for all p ≤ 599. Further, using these results, we compute the invariants [Formula: see text] and [Formula: see text] with d = -1, -3 for all p ≤ 599 and all n with the help of the computer.

scholarly journals On the diophantine equation x2 − py2 = ± 4q and the class number of real subfields of a cyclotomic field

1983 ◽  
Vol 91 ◽  
pp. 151-161 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Number Field ◽  
Diophantine Equation ◽  
Rational Number ◽  
Class Number ◽  
Cyclotomic Field ◽  
Rational Integer ◽  
Root Of Unity

Let H(m) denote the class number of the field where Q is the rational number field and ζm is a primitive m-th root of unity for a positive rational integer m.

scholarly journals On the Class Number of a Relatively Cyclic Number Field

1967 ◽  
Vol 29 ◽  
pp. 31-44 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Number Field ◽  
Class Number ◽  
Cyclotomic Field ◽  
Rational Field ◽  
Root Of Unity

Letlbe a rational prime. For eachn≧0, denote byζlna primitiveln-th root of unity and byQ(ζl,n) the cyclotomic field obtained by adjoiningζl,nto the rational field Q. Then atheoremwhich was proved by H. Weber is well known:

scholarly journals Trigonometric identities

1986 ◽  
Vol 9 (4) ◽  
pp. 705-714
Author(s):  
Malvina Baica
Keyword(s):  
Positive Integer ◽  
Cyclotomic Field ◽  
Cyclotomic Fields ◽  
Root Of Unity ◽  
Trigonometric Identities

In this paper the author obtains new trigonometric identities of the form2(p−1)(p−2)2∏k=1p−2(1−cos2πkp)p−1−k=pp−2which are derived as a result of relations in a cyclotomic fieldℛ(ρ), whereℛis the field of rationals andρis a root of unity.Those identities hold for every positive integerp≥3and any proof avoiding cyclotomic fields could be very difficult, if not insoluble. Two formulas∑k=1p−12(−1)(p2k)tanp−1−2kϕ=0  and−1+∑k=0p−12(−1)k(∑i=0p−1−2k2(p2k+2i)(k+1k))cosp−2kϕ=0stated only by Gauss in a slightly different form without a proof, are obtained and used in this paper in order to give some numeric applications of our new trigonometric identities.

scholarly journals Note on the class-number of the maximal real subfield of a cyclotomic field, II

1989 ◽  
Vol 113 ◽  
pp. 147-151
Author(s):  
Hiroyuki Osada
Keyword(s):  
Natural Number ◽  
Class Number ◽  
Class Group ◽  
Sufficient Conditions ◽  
Cyclotomic Field ◽  
Sufficient Condition ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Root Of Unity ◽  
The Ideal

For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the fieldK = Q(ζm + ζm−1)respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

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