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Über die Klassenzahl des Körpers P(ie) mit einer Primzahl p ≡ 1 mod. 2 3
... The proof of this theorem uses the arithmetic properties of the biquadratic field Q(d/-1, ~'2) of eight roots of unity. There followed a series of papers by Hasse; in [6] he proved the following theorem. Proof. ...
... Theorem 3 follows immediately. In two more papers [S, 91, Hasse used the methods of [6] and discussed the residue of h(d) (mod 8), where d has exactly two prime divisors. He treated negative discriminants in [S], and both negative and positive discriminants in [9]. ...
In this paper we study congruence conditions on class numbers of binary quadratic discriminants d, modulo powers of 2, where d has two or three distinct prime divisors.
... Cohn and Hasse showed in [BC1] and [Ha2] the congruence relations h 2 (K) 4 # p&1 8 +y mod 2 and h 2 (K) 4 # a&1 2 mod 2. ...
... By Theorem 2.1 K 2 (0 F )(2) is elementary abelian iff h(K) = 2' +1 . So the corollary is a trivial consequence of [5], [12], [13], [18]. • ...
For a number field F with ring of integers O F the tame symbols yield a surjective homomorphism with a finite kernel, which is called the tame kernel, isomorphic to K 2 ( O F ). For the relative quadratic extension E/F , where and E ≠ F , let C S (E/ F)(2) denote the 2-Sylow-subgroup of the relative S -class-group of E over F , where S consists of all infinite and dyadic primes of F , and let m be the number of dyadic primes of F , which decompose in E.
... $h(2P)\equiv 0(mod 8)$ si et seulement si $d^{2}\equiv u^{2}\equiv 1(mod 16)$ . V\'erifions que ce crit\'ere est \'equivalent \'a notre Th\'eor\'eme 1: un raisonnement \'el\'ementaire (Cf. [9], page168) montre que $(-1)^{(|e|-1)/2}=$ $(-1)^{v/2}$ , donc $(2/(a+b))=(-1)^{v/2}$ , et nous avons vu que $(2/a)=(2/d)$ . Si $a\equiv\pm 1$ et $b\equiv 0(mod 8)$ , alors $p\equiv d^{2}\equiv 1(mod 16)$ et $v^{2}\equiv 0(mod 16)$ , donc $u^{2}\equiv 1(mod 16)$ . ...
... This criterion is due to Hasse [13]. ...
Denote by the additive group of p-adic integers. The main theme of this thesis is the existence and properties of Galois extensions of algebraic number fields with Galois group , in short -extensions. We shall however also consider some non-abelian pro-p-groups as Galois groups.
We use a variant of Vinogradov’s method to show that the density of the set of prime numbers p≡-1mod4 for which the class group of the imaginary quadratic number field Q(-8p) has an element of order 16 is equal to 1/16, as predicted by the Cohen–Lenstra heuristics.
Let K be a cyclic extension of a number field k of prime degree l, f be a nonzero integral ideal of k, and ∞ be a product of certain real places of k. In this paper we study the l-Sylow subgroup L(K, f ∞) of the ideal class group of K modulo f∞. (See Section 1 for its definition.) A lower filtration of L(K, f∞) according to the action of is introduced, the cardinality of each subgroup occurring in the filtration is computed, and an algorithm leading to the structure of L(K, f∞) is given. Our work generalizes that of Morton (when l = 2) and Gras (when f is trivial).
Let h ( m ) denote the class number of the quadratic field Q (√ m ). In this paper necessary and sufficient conditions for h ( m ) to be divisible by 16 are determined when m = − p , where p is a prime congruent to 1 modulo 8, and when m = −2 p , where p is a prime congruent to ±1 modulo 8.
Sur les d'idéaux dans les extensions cycliques relatives de degré premier Annales de l'institut Fourier, tome 23, n o 4 (1973), p. 1-44. <http://www.numdam.org/item?id=AIF_1973__23_4_1_0> © Annales de l'institut Fourier, 1973, tous droits réservés. L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/), implique l'accord avec les conditions gé-nérales d'utilisation (http://www.numdam.org/legal.php). Toute utilisa-tion commerciale ou impression systématique est constitutive d'une in-fraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
Congruence conditions on the class numbers of complex quadratic fields have recently been studied by various investigators, including Barrucand and Cohn, Hasse, and the author. In this paper, we study the class number of Q(√ − pq), where p ≡ q (mod 4) are distinct primes.
P. Barrucand and H. Cohn recently gave a new criterion for the divisibility by 23 of the class number of √−p (p ≡ 1 mod 23). Here a similar criterion is given for the class number of √−2p (p ≠ 2), viz., that it is divisible by 23 iff p = ± 1 mod 23 and in an integral representation −2p = u2 − 2v2 with v>0 holds v ≡ 1, −(1+22) or 1, −1 mod 23 according to p ≡ + 1 or −1 mod 23.
We determine explicitly the quadratic subfield of a noncyclotomic Z2-extension of an imaginary quadratic number field and get a congruence property of the integer solution of a certain indeterminate equation.
Let h ( − p ) h( - p) be the class number of the quadratic field Q ( √ − p ) Q(\surd - p) , where p ≡ 1 ( mod 8 ) p \equiv 1\pmod 8 is a prime. Write p = a 2 + b 2 = 2 e 2 − d 2 p = {a^2} + {b^2} = 2{e^2} - {d^2} , where a ≡ e ≡ d ≡ b + 1 ≡ 1 ( mod 2 ) a \equiv e \equiv d \equiv b + 1 \equiv 1\pmod 2 and e > 0 e > 0 . We prove that h ( − p ) ≡ 0 h( - p) \equiv 0 or 4 ( mod 8 ) 4\pmod 8 according as ( e | p ) = 1 (e|p) = 1 or − 1 - 1 ; using this, we prove that h ( − p ) ≡ ( p − 1 ) / 2 + b ( mod 8 ) h( - p) \equiv (p - 1)/2 + b\pmod 8 . The proofs are elementary, relying on the theory of composition of binary quadratic forms.
This paper gives an elementary method to determine the number of cyclic summands with order divisible by eight in the ideal class group of a quadratic number field.
We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.
We discuss here some basic problems of algebraic number theory under a computational point of view. In particular, an explanation is given of how the application of a computer has contributed to the progress of research in this area. Several number-theoretic questions are expounded for which the use of a computer seems to be appropriate but has (apparently) not extensively been made.Wir diskutieren hier einige grundstzliche Probleme der algebraischen Zahlentheorie unter rechnrischen Gesichtspunkten. Insbesondere wird ausgefhrt, wie die Anwendung von Rechenanlagen zum Fortschritt der Forschung auf diesem Gebiet beigetragen hat. Verschiedene zahlentheoretische Fragen werden angeschnitten, fr welche die Anwendung von Rechenanlagen angemessen zu sein scheint, aber (offenbar) noch nicht intensiv praktiziert worden ist.
Let G be the separable Galois group of a finite field F of characteristic p, and X/F an imaginary hyperelliptic curve such that G acts transitively on its set W(X) of Weierstrass points. The existence of a G-invariant 2-torsion point on the Jacobian J(X) of X depends only on the parity of |W(X)|, but for large enough |F|, there exist two such curves X and X' with |W(X)|=|W(X')|, such that J(X) has (and J(X') does not have) a G-invariant 4-torsion point. The problem is equivalent to a study of the 2-,4- and 8-rank of the class number of the maximal order in the function field of such curves, and is investigated via the 2-primary class field tower. Contrary to the case of number fields, the ambiguous class depends on the discriminant, and a governing field for the 8-rank of such function fields is not known.
Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g . Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K . Let I g be the augmentation ideal of the group ring Z g over the ring of integers Z, namely I g be the ideal of Z g generated by σ − 1, σ running over all elements of g. Then is the group of all elements a σ-1 where a and σ belong to and g respectively.
Many authors have considered the divisibility of the restricted class number h+ (d) of the quadratic field Ω = Q(√d) by 4 and 8, in the case that the discriminant d of Ω, has exactly two prime factors. For such discriminants the restricted classgroup C of Ω has a nontrivial cyclic 2-Sylow subgroup, and conditions on d can be given for the existence of classes in C of orders 4 and 8. The first such results are due to Rédei. In this paper we give criteria for the divisibility of h+ (d) by 8 which are phrased in terms of the splitting of one of the prime factors p of d in a normal extension of Q depending only on d/p = d0. This simplifies and unifies the criteria for 8|h+(d) existing in the literature, which depend mainly in the representation of the prime p by certain quadratic forms, or on the quadratic character of solutions to ternary quadratic equations.
Note on Primes of Type x 2 + 32y2
- Pierre Barrucand