Quadratic Subfield Research Articles

Integral bases of dihedral numberfields. I

AbstractA dihedral number field is a non-normal quartic field K which possesses a quadratic subfield k. That is, for some integer α of k. Integral bases of these fields were known by Sommer (1907), but the form in which they were known was of little use for computational purposes. In this paper we construct integral bases of those dihedral fields with quadratic subfield of the form , d ≢ 1 (mod 8), for which only rational quantities need be determined. Although the general theory may easily be generalized to the case d ≡ 1 (mod 8), the actual determination of integral bases in this case is left to a later paper.

Zur Berechnung der Kapitulation in bikubischen Zahlk�rpern

The capitulation kernel is the kernel of the natural extension homomorphism of the ideal class groups in a extension K|k of number fields. In this paper K is a non-cyclic Galois field of degree 6 over the rationals and k is its quadratic subfield. Two different methods of computing the capitulation kernel are discussed. Both depend on the relationship between capitulation and unit structure. The paper closes with two tables. They contain the capitulation kernel for all ramified extensions K|k having cubic discriminants between −20000 and 100000.

The construction of unramified cyclic quartic extensions of 𝑄(√𝑚)

We give an elementary general method for constructing fields K satisfying [ K : Q ] = 8 [K:Q] = 8 , the Galois group of K over Q is dihedral, and K is unramified over one of its quadratic subfields. Given an integer m, we describe all such fields K which contain Q ( m ) Q(\sqrt m ) . The description is specific and is given in terms of the arithmetic of the quadratic subfields of K.

Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff K = Q((rd + p d 1 2 ) 1 2 ) , where d > 1, p and r are rational integers, d squarefree, for which p 2 + q 2 = r 2 d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, d( K Q ) , of the general cyclic quartic extension is given by d( K Q ) = (W 2d 2) for an explicitly computable rational integer W. We next find that the relative discriminant, d( K F ) , is given by d( K F ) = (W d 1 2 ) , where F = Q (d 1 2 ) is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of F = Q(d 1 2 ) is odd then every cyclic quartic extension K of Q containing F has a relative integral basis over F. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic field F is contained in some cyclic quartic extension of Q and suppose that F has even (wide) class number. There then is a cyclic quartic extension K of Q containing F such that K has no relative integral basis over F.

The quadratic and quartic character of certain quadratic units. II

Since then a number of similar results have been obtained for certain other quadratic and quartic symbols using such tools as cyclotomy, rational biquadratic reciprocity laws, etc. (see Brandler [2], Lehmer [4], [5], [6]). In this paper we apply the ideas of Barrucand and Cohn [1] in other biquadratic fields with unique factorization, thereby reproving some known results, proving some conjectures of E. Lehmer [6] and obtaining some additional new results. The method succeeds in the 21 imaginary bicyclic biquadratic fields having class number 1 and which contain QiV—ϊ), QCl/^) or ζ>(τ/~2) as a subfield. It would be interesting to know if similar techniques can be used in the remaining 26 imaginary bicyclic biquadratic fields with class number 1 or to determine octic symbols. (For a complete list of the imaginary bicyclic biquadratic fields with class number 1 see Brown and Parry [3].) We now sketch the method used. First the quadratic or quartic symbol under consideration is expressed in terms of the representation of p by the indefinite form associated with the real quadratic subfield of the biquadratic field. This is accomplished using Jacobi's form of the law of quadratic reciprocity, and the results are given in the table below. In the case of those results involving quartic symbols it is first necessary to observe that 2em is a square in the quadratic subfield, and this brings in the symbol (2/p)4 whose value is well known, viz., if p = 8n + 1 is prime so that p — a + 16b = c + 8d then (2/p)4 = (-I) 6 = (-l) Λ + d . Next we consider a prime

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are Q (2 1/2), Q ( q 1/2), and Q (2 q) 1/2 ( q prime). A special role belongs to q = A 2 + 32 b 2.

Classes et unités des extensions cycliques réelles de degré 4 de ${\bf Q}$

Let K be a real cyclic extension of degree 4 of Q ; let k be its quadratic subfield. We determine the class number and the units of K, and we prove that the problem of the “capitulation” of classes of k in K is characterized by elementary properties of the units of K. We have obtained a numerical table of the class number, the units, and the eventual “capitulation” of a class, for all fields K with conductor f<4000 ; we publish here an extract of it.

Integral bases for bicyclic biquadratic fields over quadratic subfields

A classical question of algebraic number theory is, does an algebraic number field K have an integral basis over a subfield fc? A complete and explicit answer to the above question is given here when K is a bicyclic biquadratic number field and k is a quadratic subfield. Moreover, an explicit integral basis is given for K/k whenever one exists. In the cases where k is imaginary or k is real and has a unit of norm - 1, the conditions involve only rational congruences. When k is real and the fundamental unit of e has norm + 1, the conditions sometimes involve e.

Zero divisors in tensor products of division algebras

Theorem. If a tensor product of a division algebra D D with a quaternion algebra Q Q is not a division algebra, then either D D and Q Q possess a common quadratic subfield or D D contains a splitting field of Q Q not quadratic over the base field. The above theorem generalizes a recently published result of Albert’s. Theorem. If the tensor product of two division algebras over a local or a global field K K is not a division algebra, then they contain a common extension field of K K .

Tensor Products of Quaternion Algebras

Two quaternion division algebras have a common quadratic subfield if their tensor product contains zero-divisors.

Tensor products of quaternion algebras

Two quaternion division algebras have a common quadratic subfield if their tensor product contains zero-divisors.

The Quadratic Subfield of the Field Generated by the p-th Root of Unity

The Quadratic Subfield of the Field Generated by the p-th Root of Unity

On totally complex quartic fields with small discriminants

A table of totally complex quartic fields, with small discriminants, which possess quadratic subfields was given by Delone and Faddeev(1); the purpose of the present paper is to give a table of such fields which have no subfield. The method used is based on an idea used recently (2) in the case of totally real quartic fields and depends on a special case of the following theorem, which is stated in full generality.

The quadratic subfields of a generalized quaternion algebra

The quadratic subfields of a generalized quaternion algebra