Quartic Fields Research Articles

On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

Let L \mathcal {L} be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in L \mathcal {L} is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided. These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime p ≡ 1 ( mod 5 ) p \equiv 1\;\pmod 5 . Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in O ( ( log ⁡ p ) 3 ) O({(\log p)^3}) binary operations.

On Principal Ideal Testing in Totally Complex Quartic Fields and the Determination of Certain Cyclotomic Constants

Let $\mathcal {L}$ be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in $\mathcal {L}$ is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided. These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime $p \equiv 1\;\pmod 5$. Given any quadratic (or quintic) nonresidue of p, it is shown that these cyclotomic numbers can be efficiently computed in $O({(\log p)^3})$ binary operations.

Class numbers of the simplest cubic fields

Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by n, for any n. Using 2-descents on elliptic curves, we obtain precise information on the 2-Sylow subgroups of the class groups of these fields. A theorem of H. Heilbronn associates a set of quartic fields to the class group. We show how to obtain these fields via these elliptic curves.

Calculation of the class numbers of imaginary cyclic quartic fields

Any imaginary cyclic quartic field can be expressed uniquely in the form K = Q ( A ( D + B D ) ) K = Q(\sqrt {A(D + B\sqrt D )} ) , where A is squarefree, odd and negative, D = B 2 + C 2 D = {B^2} + {C^2} is squarefree, B > 0 , C > 0 B > 0,C > 0 , and ( A , D ) = 1 (A,D) = 1 . Explicit formulae for the discriminant and conductor of K are given in terms of A, B, C, D. The calculation of tables of the class numbers h ( K ) h(K) of such fields K is described.

A Remark on a Theorem of W. E. H. Berwick

A Remark on a Theorem of W. E. H. Berwick

On class numbers of cyclic quartic fields

On class numbers of cyclic quartic fields

Finding fundamental units in algebraic number fields

Recently Cusick [4] presented a very elegant and short proof of the fact that a pair of fundamental units of a totally real cubic or quartic number field can be found by taking ‘successive minima’ of the function tr(ε2), where ε runs through the group of units and tr denotes the absolute trace. Hidden in Cusick's proof there is a general theorem, which shows how strictly convex functions can be used to find lattice-vectors extensible to a basis of a given geometrical lattice, and which we state and prove in § 3. A result analogous to Cusick's for some families of functions related to tr (ε2) is given in Theorem 1, thereby improving results of Brunotte and Halter-Koch [2], [5]. For a survey of unit groups of rank 2 and more literature the reader is also referred to [2].

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.

A generalization of Voronoi's unit algorithm II

It is proved that the generalized Voronoi algorithm developed in Part I of this paper computes the fundamental units of all fields of unit rank 2, i.e., of the totally real cubic fields, of the quartic fields with two real conjugate fields, of the quintic fields with only one real conjugate field, and of the totally complex sextic fields.

A generalization of Voronoi's unit algorithm I

By means of a new method of mapping an algebraic number field into a euclidean space Voronoi's unit algorithm is generalized to all algebraic number fields and it is proved that the generalized Voronoi algorithm computes the fundamental units of all algebraic number fields of unit rank 1, i.e., of the real quadratic fields, of the complex cubic fields, and of the totally complex quartic fields.

Integral generators in a certain quartic field and related Diophantine equations.

Integral generators in a certain quartic field and related Diophantine equations.

Class number calculation of a quartic field from the elliptic unit

Class number calculation of a quartic field from the elliptic unit

Extension of a theorem of Cauchy and Jacobi

Let q and p be prime with q = a 2 + b 2 ≡ 1 (mod 4), a ≡ 1 (mod 4), and p = qf + 1. In the nineteenth century Cauchy ( Mém. Inst. France 17 (1840) , 249–768) and Jacobi ( J. für Math. 30 (1846) , 166–182) generalized the work of earlier authors, who had determined certain binomial coefficients (mod p) (see H. J. S. Smith, “Report on the Theory of Numbers,” Chelsea, 1964), by determining two products of factorials given by Π k kf! (mod p = qf + 1) where k runs through the quadratic residues and the quadratic non-residues (mod q), respectively. These determinations are given in terms of parameters in representations of p h or of 4 p h by binary quadratic forms. A remarkable feature of these results is the fact that the exponent h coincides with the class number of the related quadratic field. In this paper C. R. Mathews' ( Invent. Math. 54 (1979) , 23–52) recent explicit evaluation of the quartic Gauss sum is used to determine four products of factorials (mod p = qf + 1, q ≡ 5 (mod 8) > 5), given by Π k kf! where k runs through the quartic residues (mod q) and the three cosets which may be formed with respect to this subgroup. These determinations appear to be considerably more difficult. They are given in terms of parameters in representations of 16 p h by quaternary quadratic forms. Stickelberger's theorem is required to determine the exponent h which is shown to be closely related to the class number of the imaginary quartic field Q( i√2 q + 2 a√ q), q = a 2 + b 2 ≡ 5 (mod 8), a odd.

Class numbers of imaginary cyclic quartic fields and related quaternary systems

Class numbers of imaginary cyclic quartic fields and related quaternary systems

Remark on the Divisibility of the Class Numbers of Certain Quartic Numbers Fields by 5

Remark on the Divisibility of the Class Numbers of Certain Quartic Numbers Fields by 5

Cyclic quartic fields and genus theory of their subfields

Let k = Q (√ u) ( u ≠ 1 squarefree), K any possible cyclic quartic field containing k. A close relation is established between K and the genus group of k. In particular: (1) Each K can be written uniquely as K = Q (√ vwη), where η is fixed in k and satisfies η ⪢ 1, (η) = U 2√ u, | U 2| = |(√ u)|, ( v, u) = 1, v ∈ Z is squarefree, w| u, 0 < w < √ u. Thus if u ≠ a 2 + b 2, there is no K ⊃ k. If u = a 2 + b 2 then for each fixed v there are 2 g − 1 K ⊃ k, where g is the number of prime divisors of u. (2) K k has a relative integral basis (RIB) (i.e., O K is free over O k ) iff N( ε 0) = −1 and w = 1, where ε 0 is the fundamental unit of k, (or, equivalently, iff K = Q (√ vε 0√ u), ( v, u) = 1). (3) A RIB is constructed explicitly whenever it exists. (4) disc( K) is given. In particular, the following results are special cases of (2): (i) Narkiewicz showed in 1974 that K k has a RIB if u is a prime; (ii) Edgar and Peterson ( J. Number Theory 12 (1980), 77–83) showed that for u composite there is at least one K ⊃ k having no RIB. Besides, it follows from (4) that the classification and integral basis of K given by Albert ( Ann. of Math. 31 (1930), 381–418) are wrong.

On quartic fields of signature one with small discriminant. II

Tables of quartic fields, having two real and two complex conjugates, and with discriminants between − 7776 -7776 and 0, are given.

On a sequence arising in series for 𝜋

In a recent investigation of dihedral quartic fields [6] a rational sequence { a n } \{ {a_n}\} was encountered. We show that these a n {a_n} are positive integers and that they satisfy surprising congruences modulo a prime p. They generate unknown p-adic numbers and may therefore be compared with the cubic recurrences in [1], where the corresponding p-adic numbers are known completely [2]. Other unsolved problems are presented. The growth of the a n {a_n} is examined and a new algorithm for computing a n {a_n} is given. An appendix by D. Zagier, which carries the investigation further, is added.

Corrigenda: “On quartic fields of signature one with small discriminant. II”

Corrigenda: “On quartic fields of signature one with small discriminant. II”

Products and sums of powers of binomial coefficients mod 𝑝 and solutions of certain quaternary Diophantine systems

In this paper we prove that certain products and sums of powers of binomial coefficients modulo p = q f + 1 p = qf + 1 , q = a 2 + b 2 q = {a^2} + {b^2} , are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition \[ 16 p α = x 2 + 2 q u 2 + 2 q v 2 + q w 2 , ( x , u , v , w , p ) = 1 , x w = a v 2 − 2 b u v − a u 2 , x ≡ 4 ( mod q ) , α ⩾ 1. \begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array} \] The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field \[ K = Q ( i 2 q + 2 a q ) , K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ), \] as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field Q ( e 2 π i p / q ) Q({e^{2\pi ip/q}}) . Let the four cosets of the subgroup A of quartic residues be given by c j = 2 j A , j = 0 , 1 , 2 , 3 {c_j} = {2^j}A,j = 0,1,2,3 , and let \[ s j = 1 q ∑ t ∈ c j t , j = 0 , 1 , 2 , 3. {s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.} \] Let s m {s_m} and s n {s_n} denote the smallest and next smallest of the s j {s_j} respectively. We give new, and unexpectedly simple determinations of Π k ∈ c n k f ! {\Pi _{k \in {c_n}}}kf! and Π k ∈ c n + 2 k f ! {\Pi _{k \in {c_{n + 2}}}}kf! , in terms of the parameters x in the above partition of 16 p α 16{p^\alpha } , in the complicated case that arises when the class number of K is &gt; 1 &gt; 1 and s m ≠ s n {s_m} \ne {s_n} .