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The determination of the imaginary abelian number fields with class number one
Abstract
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, √-163).
... The presently best known bounds are due to Lee-Kwon [19], who in 2006 showed that d ≤ 216 (and d ≤ 96 assuming GRH). For abelian CM-fields the class number one problem was solved in 1994 by Yamamura [43] and the relative class number one problem in 2000 by Chang-Kwon [6]. Since 1994, solving the class number one problem in the non-abelian case for specific degrees and Galois groups has been a major undertaking by various authors. ...
... Abelian fields. The abelian CM-fields with class number one were determined by Yamamura in [43]. There are 172 such fields, the largest field having degree 24. ...
We show that assuming the generalized Riemann hypothesis there are no normal CM-fields with class number one of degree 64 and 96. This is done by constructing complete tables of normal CM-fields using discriminant bounds of Lee--Kwon. This solves the class number one problem for normal CM-fields assuming GRH. Using the same technique to solve the relative class number one problem in degrees 16, 32, 56 and 82, also the corresponding relative class number one problem is solved assuming GRH.
... is an imaginary abelian extension of Q, and from Yamamura [28], we know the imaginary abelian number fields with class number 1. ...
... Taking K general and looking at the Hilbert class field of K , we conclude that h K can be 1 or 2. We cannot say that h K = 1, like the C 2 case: for example, we know from [28] ...
A number field K is Hilbert–Speiser if all of its tame abelian extensions L / K admit NIB (normal integral basis). It is known that is the only such field, but when we restrict to be a given group G, the classification of G-Hilbert–Speiser fields is far from complete. In this paper, we present new results on so-called G-Leopoldt fields. In their definition, NIB is replaced by “weak NIB” (defined below). Most of our results are negative, in the sense that they strongly limit the class of G-Leopoldt fields for some particular groups G, sometimes even leading to an exhaustive list of such fields or at least to a finiteness result. In particular, we are able to correct a small oversight in a recent article by Ichimura concerning Hilbert–Speiser fields.
... When K is an abelian extension K( √ −3) is an imaginary abelian extension of Q, and from Yamamura [Yam94] we know them all. Remark 3.4. ...
... Remark 3.4. Taking K general, looking at the Hilbert class field of K we conclude that h K can be 1 or 2. We cannot say that h K = 1, like the C 2 case: for example we know from [Yam94] ...
In the last decades a lot of work has been done in the study of Hilbert-Speiser fields. The first input came from the important result contained in "Swan modules and Hilbert-Speiser number fields" (Greither, Replogle, Rubin and Srivastav): is the only Hilbert-Speiser field, i.e. for every number field there exists a (cyclic of prime order) tame abelian extension that does not have NIB. Then the research went towards the finer problem of finding crieria for -Hilbert-Speiser fields, i.e. fields such that every tame extension with Galois group isomorphic to , the cyclic group of prime order l, has NIB. In this work our purpose is to consider a weakened version of NIB: we will say that a tame abelian extension L/K of number fields has a weak normal integral basis if is free of rank 1 over , where is the maximal order of K[G]. WNIB's have been studied for instance in [Gre90], [Gre97] and [GJ12]. We shall ask the same questions substituting "WNIB" to "NIB" everywhere, and we will denote the analogous condition of "Hilbert-Speiser field" by "Leopoldt field". We are going to find mainly necessary conditions for number fields to be -Leopoldt, where as before l is a prime number, which also give criteria and (sometimes conditional) finiteness results for -Hilbert-Speiser fields; for instance we will see that this permits to correct a mistake contained in the article "Real abelian fields satisfying the Hilbert-Speiser condition for some small primes p" by Ichimura, whose overall Hilbert-Speiser-oriented techniques are supple enough to be applied to our problem.
... Brown and Parry [5] determine a complete list of imaginary biquadratic fields with class number 1. Yamamura [18] gives a complete list of imaginary abelian fields with class number 1. Jung and Kwon [10] show a complete list of imaginary biquadratic fields of class number 3. Elsenhans, Klüners and Nicolae [6, Theorem 1] present a complete list of imaginary quadratic fields with class groups of exponent E for every E ≤ 5 and E = 8 under the extended Riemann hypothesis (ERH). We say that K is an n-quadratic field if K is a Galois extension of Q with Gal(K/Q) ≃ C n 2 where C 2 is the cyclic group of order 2. In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5, that is, isomorphic to the direct products C r u of the cyclic group C u of order u with u = 3, 5 and positive integers r, under ERH for every positive integer n. ...
In this paper we obtain a complete list of imaginary n-quadratic fields with class groups of exponent 3 and 5 under ERH for every positive integer n where an n-quadratic field is a number field of degree represented as the composite of n quadratic fields.
... where a 1 and a 2 are given in the following [8] and Yamamura [18] independently determined the imaginary triquadratic fields of class number 1. Feaver further proved that for n > 3 there are no imaginary n-quadratic fields of class number 1. ...
This paper gives a method to find all imaginary multiquadratic fields of class number dividing provided the list of all imaginary quadratic fields of class number dividing is known. We give a bound on the degree of such fields. As an application of this algorithm, we compute a complete list of imaginary multiquadratic fields with class number dividing $32.
... Therefore, in order to find super-isolated surfaces of near-prime order (note that near-prime order implies the hypothesis in Corollary 19), it is sufficient to find all super-isolated Weil numbers π, such that ππ is prime and Q(π) has degree 4. There are 91 quartic CM fields of class number 1, and they can be found in the literature [13,30]. By [14,Cor. ...
We call a simple abelian variety over super-isolated if its (-rational) isogeny class contains no other varieties. The motivation for considering these varieties comes from concerns about isogeny based attacks on the discrete log problem. We heuristically estimate that the number of super-isolated elliptic curves over with prime order and , is roughly . In contrast, we prove that there are only 2 super-isolated surfaces of cryptographic size and near-prime order.
... The ideal class number one problem has its origin in the Disquisitiones Arithmeticae of F. Gauss (1807). For quadratic imaginary number fields, the problem was solved in 1966 by H. Stark and for general Abelian imaginary number fields, it was solved in 1994 by K. Yamamura [12]: there are 172 imaginary Abelian number fields with class number one. The ideal class number one problem has an analogue for function field extensions KÂk, where k=F q (x) with some element x transcendental over F q . ...
In this paper, we determine all finite separable imaginary extensions K/Fq(x) whose maximal order is a principal ideal domain in case K/Fq(x) is a non zero genus cyclic extension of prime power degree. There exist exactly 42 such extensions, among which 7 are non isomorphic over Fq.
... Moreover, if we assume the Generalized Riemann Hypothesis, then any normal CM-field with class number one is of degree less than or equal to 96 ( [2] and [18]). All imaginary abelian number fields with relative class number one are known: their degrees are less than or equal to 24 ([7] and [51]). All normal CM-fields of degrees less than 48 and class number one are known. ...
It is known that if we assume the Generalized Riemann Hypothesis, then any normal CM-field with relative class number one is of degree less than or equal to 96. All normal CM-fields of degree less than 48 with class number one are known. In addition, for normal CM-fields of degree 48 the class number one problem is partially solved. In this paper we will show that under the Generalized Riemann Hypothesis there is no more normal CM-fields with class number one except for the possible fields of degrees 64 or 96.
... Uchida [17,Theorem 2] proved that there exist only finitely many imaginary abelian number fields with class number one. In fact, the class number one problem for imaginary abelian number fields has lately been settled by Yamamura [20], and it is now known that there are exactly fifty-four imaginary abelian quartic number fields with class number one, that forty-seven of them are bicyclic biquadratic (see [1]) and that seven of them are cyclic quartic (see [14]). Hence, it is time to move on to the determination of the non-abelian or even non-normal CM-fields with class number one since Uchida [17,Remark 1] also proved that there exist only finitely many CM-fields of fixed degree with class number one. ...
We give explicit upper bounds for the discriminants of the non-normal quartic CM-fields with class number one, and forthe discriminants of the dihedral octic CM-fields with class number one. These upper bounds are too large to enable usto achieve the determination of these number fields. Nevertheless, whenever a real quadratic number field kis fixed, we can explain how to determine the non-normal quartic CM fields or the dihedral octic CM-fields with class number one and with real quadratic subfield k. © 1994, Mathematical Institute, Tohoku University. All rights reserved.
Let r>2 and l be primes. In this paper we study the mod l Galois representations attached to curves of the form y^r = f(x) where f is monic and has coefficients belonging to the r-th cyclotomic field. We provide conditions on the coefficients (and degree) of f which allow one to verify the mod l image is large outside of a (typically small) finite explicit set of primes. We allow all values of r for which the r-th cyclotomic field has odd class number. This appears to be the first explicit result for abelian varieties of dimension greater than two and not of GL_2-type which allows the ground field to have unramified extensions. In proving the large image result we give a classification of the maximal subgroups containing transvections of certain classical groups and describe (in many cases) the images of inertia groups. The exact mod l image is governed by the "endomorphism character", a certain algebraic Hecke character which generalises the CM character. When r=3, we depict the image in its entirety. To the author's knowledge, this is the first accurate description in the literature. Finally, we give several examples with genus ranging from 10 to 36. Applications to the Inverse Galois Problem are also included.
We call an abelian variety over a finite field Fq super-isolated if its (Fq-rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g≥3, there are finitely many such varieties over all finite fields.
In this paper, we present a complete classification of all imaginary n-quadratic fields of class number 1.
For a prime number p, we say that a number field F satisfies the Hilbert-Speiser condition (Hp) if each tame cyclic extension N=F of degree p has a normal integral basis. In this note, we determine the real abelian number fields satisfying (Hp) for odd prime numbers p with h(Q(√-p))= 1.
We shall deal with infinite towers of cyclic fields of genus number 1 in which a prime number l ≥ 5 is totally ramified. Our main result states that, if m is a positive divisor of l - 1 less than (l - 1)/2, then for any positive integer n, the cyclic field of degree mln with conductor ln+1 is not norm-Euclidean. In particular, it follows that, for any positive integer n, the (real) cyclic field of degree ln with conductor ln+1 is not norm-Euclidean and that the (imaginary) cyclic field of degree 14 with conductor 49, whose class number is known to equal 1, is not norm- Euclidean.
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.
We provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at s = 1 of L-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for CM-fields, we deduce an upper bound for the root discriminants of the normal CM-fields with (relative) class number one.
The last chapter brings a short survey on the work on Fermat’s Last Theorem, culminating in its proof.
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. It is known that there are only finitely many CM-fields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Second, we have proved in this paper that there are exactly 48 such fields.
To date, the class number one problem for non-normal CM-fields is solved only for quartic CM-fields. Here, we solve it for a family of non-normal CM-fields of degree 2p, p⩾3p⩾3 and odd prime. We determine all the non-isomorphic non-normal CM-fields of degree 2p, containing a real cyclic field of degree p, and of class number one. Here, p⩾3p⩾3 ranges over the odd primes. There are 24 such non-isomorphic number fields: 19 of them are of degree 6 and 5 of them are of degree 10. We also construct 19 non-isomorphic non-normal CM-fields of degree 12 and of class number one, and 10 non-isomorphic non-normal CM-fields of degree 20 and of class number one.
In this note, we apply the method of Rodriguez Villegas and Yang (1996) to construct a family of innite many theta series over the Hilbert- Blumenthal modular surfaces with a common zero. We also relate the non- vanishing of the central L-values of certain Hecke characters of non-biquadratic CM number elds of degree 4 to the nonvanishing of theta functions at CM points in the Hilbert-Blumenthal modular surfaces.
We prove that there are only finitely many CM-fields N with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of N. Since their actual determination would be too difficult, we only content ourselves with the determination of the nonquadratic imaginary cyclic number fields of 2-power degrees with cyclic ideal class groups of 2-power orders. There are exactly 22 such number fields, 10 of them having class number one, 9 of them having class number two, and 3 of them having class number four. This present determination is a nice complement to the determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ⩽2 completed in an earlier paper.
Let p be a prime number. We say that a number field F satisfies the condition when any tame cyclic extension N/F of degree p has a normal integral basis. We determine all the CM Galois extensions F/\mbi{Q} satisfying for the case , using the action of the complex conjugation on several objects associated to F .
We prove that there are 95 non-isomorphic totally complex quartic fields whose rings of algebraic integers are generated by an algebraic unit and whose class numbers are equal to 1. Moreover, we prove Louboutin's Conjecture according to which a totally complex quartic unit εu generally generates the unit group of the quartic order Z[εu].
Lately, I. Miyada proved that there are only finitely many imaginary abelian number fields with Galois groups of exponents
≤2 with one class in each genus. He also proved that under the assumption of the Riemann hypothesis there are exactly 301
such number fields. Here, we prove the following finiteness theorem: there are only finitely many imaginary abelian number
fields with one class in each genus. We note that our proof would make it possible to find an explict upper bound on the discriminants
of these number fields which are neither quadratic nor biquadratic bicyclic. However, we do not go into any explicit determination.
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).
In this paper we determine all the abelian extensions of rational function fields which have divisor class number one. We also determine all the imaginary abelian extensions with relative divisor class number one.
We prove that the relative class number of a nonabelian normal CM-field of degree 2pq (where p and q are two distinct odd primes) is always greater than four. Not only does this result solve the class number one problem for the nonabelian normal CM-fields of degree 42, but it has also been used elsewhere to solve the class number one problem for the nonabelian normal CM-fields of degree 84.
We give a necessary condition for the ideal class group of a CM-field to be of exponent at most two. This condition enables us to drastically reduce the amount of relative class number computation for determination of the CM - fields of some types (e. g. the imaginary cyclic non -quadratic number fields of 2 - power degrees) whose ideal class groups are of exponents at most two. We also give a necessary condition for some quartic non - CM - fields to have class number one.
We show that there exist only finitely many imaginary abelian number fields of type (2,2,...,2) with one class in each genus.
Moreover, if the Generalized Riemann Hypothesis is true, we have exactly 301 such fields, whose degrees are less than or equal
to 23. Finally we give the table of those 301 fields.
Let K be a number field and let G be a finite abelian group. We call
K a Hilbert-Speiser field of type G if, and only if, every tamely
ramified normal extension L/K with Galois group isomorphic to G has
a normal integral basis. Now let C2 and C3 denote the cyclic
groups of order 2 and 3, respectively. Firstly, we show that among all
imaginary quadratic fields, there are exactly three Hilbert-Speiser
fields of type
C2: \mathbbQ(Öm)C_{2}: \mathbb{Q}(\sqrt {m}), where m Î {-1, -3, -7}m \in \{-1, -3, -7\}.
Secondly, we give some necessary and sufficient conditions for a real
quadratic field
K = \mathbbQ(Öm)K = \mathbb{Q}(\sqrt {m}) to be a Hilbert-Speiser field of
type C2. These conditions are in terms of the congruence class of m
modulo 4 or 8, the fundamental unit of K, and the class number of K.
Finally, we show that among all quadratic number fields, there are
exactly eight Hilbert-Speiser fields of type
C3: \mathbbQ(Öm)C_{3}: \mathbb{Q}(\sqrt {m}),
where m Î {-11,-3, -2, 2, 5, 17, 41, 89}m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}.
We give explicit upper bounds for residues at s=1 of Dedekind zeta functions of number fields, for |L(1, χ)| for nontrivial primitive characters χ on ray class groups, and for relative class numbers of CM fields. We also give explicit lower bounds for relative class numbers of CM fields (which do not contain any imaginary quadratic subfield). Most of these bounds stem from a useful lemma which enables us to easily obtain neat bounds.
We show that the normal CM-fields with relative class number one are of degrees ≤216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CM-fields with relative class number one are of degrees ≤96, and the CM-fields with class number one are of degrees ≤104. By many authors all normal CM-fields of degrees ≤96 with class number one are known except for the possible fields of degree 64 or 96. Consequently the class number one problem for normal CM-fields is solved under the Generalized Riemann Hypothesis except for these two cases.
We prove that there is precisely one normal CM-field of degree 48 with class number one which has a normal CM=subfield of degree 16: the narrow Hilbert class field of Q(√5, √101, θ) with θ3 - θ2 -5θ - 1 = 0.
Associative division algebras are a rich source of fully diverse space-time
block codes (STBCs). In this paper the systematic construction of fully diverse
STBCs from nonassociative algebras is discussed. As examples, families of fully
diverse , multiblock and 4\x 4 STBCs are designed,
employing nonassociative quaternion division algebras.
Let p be an odd prime number, K a CM-field, and K-infinity/K the cyclotomic Z(p)-extension with its n-th layer K-n (n greater than or equal to 0). Let A(n) be the Sylow p-subgroup of the ideal class group of K-n. For the odd part A(n)(-) of A(n)(-), it is well known that the natural map A(n)(-) --> A(n+1)(-) is injective. The purpose of this note is to show that an analogous phenomenon occurs for the Galois module structure of rings of integers of a certain class of tamely ramified extensions over K-n of degree p.
The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, "No doubt the same method will work for other small class numbers, up to 100, say." However, more than ten years later, after doing W = 5,6,7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large "exceptional modulus" of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.
[Kea87, Las86]. Abundance [BH86]. abundant [Coh84]. Academic [Ano88b, Car80, Che83, Fet81e, Fet82a, Kru83, Sol86, vK82]. Accelerated [Hug86, AH81, Had82, HH88b, Mar80c]. accelerating [AJS83, Lev80]. Acceleration [Bre82b, Kel86, Saa84, GL82]. accelerators [SF82]. Accuracy [JS85, Mon85a, BM82b, Eri85c, GL85, Sch80b]. Accurate [Gar84, HH88a, MNP84, BDK82, BSC81, DK85, HTE81, OS86b, San88, Sch83a]. adapted [Eri85c]. Adaptive [VB81c, EJ88, MT86, OO84]. Addendum [Fra81a, Zag88]. addition [McC86]. Additive [CS80a, CS83a, LO83]. ADI [HV89]. adic [Dea85]. Admitting [For87]. advanced [Gre82]. Advection [Pas80, Hal86]. Aerospace [Fet81b]. ahead [PTL85]. Air [Fet81b]. Aitken [MP85]. Algebraic [Boy85a, Boy85b, BE88b, Smy84, AD88, BE88a, Bru82, BW88, BW89, Eas86, GN88, Ger83a, KLL88, LP86, Rhe84, Sid80b, Smy81, Toi86, Vau83, Vau84, Vau85b]. Algorithm [AL81, Ger88, GM84, HJ81, HW84, PSS82, Sto80, Yan87, Bai88a, BF89a, BC82b, CZ81, Cas80, DL88b, EK89, Fie81, FM85, For81, Gar88a, Ger83c, ...
Let K {\mathbf {K}} be a CM-field that is a quadratic extension of a totally real number field k {\mathbf {k}} . Under a technical assumption, we show that the relative class number of K {\mathbf {K}} is large compared with the absolute value of the discriminant of K {\mathbf {K}} , provided that the Dedekind zeta function of k {\mathbf {k}} has a real zero s s such that 0 > s > 1 0 > s > 1 . This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.
In this paper we describe the calculation of the class numbers of most real abelian number fields of conductor . The technique is due to J. M. Masley and makes use of discriminant bounds of A. M. Odlyzko. In several cases we have to assume the generalized Riemann hypothesis.
For a number of years, French mathematicians have run regular number theory conferences to which they have invited number theorists from many countries. To repay their hospitality, the London Mathematical Society arranged for the 1980 'Journees Arithmeiques' to be held in Exeter. The papers published here are either based on the main invited lectures or on selected research talks given at the conference. They cover all branches of the subject: combinatorial and elementary methods; analytic number theory; transcendence theory; Galois module theory and algebraic number theory in general; elliptic curves and modular functions; local fields; additive number theory; Diophantine geometry, and uniform distribution. It will be necessary reading for all those undertaking research in number theory.
Real cyclic cubic fields.- Real cyclic sextic fields.- The function ? and the structure of UR.- Bergstrom's product formula.- Bergstrom's product formula in the case of real cyclic sextic fields.- Formulas for computing ?A.- The class number of K6.- The signature rank of U6.- The computer program.- Numerical results.
Let K be a finite normal extension of an algebraic number field k; let k 2 be the compositum of all quadratic extensions of k which are contained in K. Let ζ k (s), ζ K (s) and ζ k2 (s) denote the Dedekind ζ-functions of these fields.
The authors propose a hypothesis whose proof would provide, in particular, a solution of the "class-number 1" problem, recently solved by Stark and Baker.
If [chi](n) is an odd real character (mod k), it is an unsolved problem whether . We propose a hypothesis which ensures the truth of this inequality.
Lecture Note Ser. 56
- London Math
- Soc
London Math. Soc. Lecture Note Ser. 56, Cambridge Univ. Press, Cambridge, New York,
1982, pp. 151-193.
+a\X"~x h-va " giving an unramified A " -extension of a real quadratic number field Q{VD) with class number one (unpublished tables). 34. _, Some analogue of Hubert's irreducibility theorem and the distribution of algebraic number fields
_,
Examples of polynomials X"+a\X"~x h-va "
giving an unramified A " -extension
of a real quadratic number field Q{VD) with class number one (unpublished tables).
34. _,
Some analogue of Hubert's irreducibility theorem and the distribution of algebraic
number fields, J. Fac. Sei. Univ. Tokyo Sect. IA 38 (1991), 99-135.
On real characters 212-213. 8. _, On real zeros ofDedekind (..functions, Cañad
- H Heilbronn
H. Heilbronn, On real characters, Acta Arith. 2 (1937), 212-213.
8. _,
On real zeros ofDedekind (..functions, Cañad. J. Math. 25 (1973), 870-873.
- Wiss
- Berlin
Wiss. Berlin, Kl. Math. Phys. 2 (1964), 1-64.
Algebraic Number Fields: L functions and Galois Properties
- J M Masley
J. M. Masley, Odlyzko bounds and class number problems, Algebraic Number Fields: L functions and Galois Properties (A. Fröhlich, ed.), (Proc. Sympos. Univ., Durham, 1975),
Academic Press, London, 1977, pp. 465-474.
On real zeros ofDedekind (..functions, Cañad
_,
On real zeros ofDedekind (..functions, Cañad. J. Math. 25 (1973), 870-873.
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27. H. M. Stark, Some effective versions of the Brauer-Siegel theorem, Invent. Math. 23 (1974),
135-152.
Real roots of real Dirichlet 𝐿-series
- Rosser, J. Barkley