Cyclic Quartic Fields Research Articles

Determination of all imaginary cyclic quartic fields of prime class number p ≡ 3(mod4), and non-divisibility of class numbers

Let [Formula: see text] be a prime such that [Formula: see text]. Then, we show that there is no imaginary cyclic quartic extension [Formula: see text] of [Formula: see text] whose class number is [Formula: see text]. Suppose [Formula: see text] is a cyclic extension of number fields with an odd degree. Then, we show that [Formula: see text] does not divide the class number of [Formula: see text] if the class group of [Formula: see text] is cyclic. We also construct some families of number fields whose class number is not divisible by a fixed prime.

Well-Rounded ideal lattices of cyclic cubic and quartic fields

In this paper, we find criteria for when cyclic cubic and cyclic quartic fields have well-rounded ideal lattices. We show that every cyclic cubic field has at least one well-rounded ideal. We also prove that there exist families of cyclic quartic fields which have well-rounded ideals and explicitly construct their minimal bases. In addition, for a given prime number $p$, if a cyclic quartic field has a unique prime ideal above $p$, then we provide the necessary and sufficient conditions for that ideal to be well-rounded. Moreover, in cyclic quartic fields, we provide the prime decomposition of all odd prime numbers and construct an explicit integral basis for every prime ideal.

Diophantine problems related to cyclic cubic and quartic fields

We are interested in solving the congruences f3+g3+1≡0(modfg) and f4−4g2+4≡0(modfg) in polynomials f,g with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively. Our approach is based on Gröbner basis techniques and we do numerical experiences based on it. We mainly deal with the case degf≤2 and prove that there are no new families of cyclic cubic nor cyclic quartic fields. In case of cyclic quartic fields we obtained a new polynomial that was not discovered by Balady and Washington [2], it is given by t4+7890798742t3−37333446t2+38618t+1.

The construction of the Hilbert genus fields of real cyclic quartic fields

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. In the present paper, we construct the Hilbert genus field of the real cyclic quartic fields $\mathbb{Q}(\sqrt{a\varepsilon_p\sqrt{p}})$.

Euclidean algorithm in Galois quartic fields

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number 1 are Euclidean.

On Hilbert genus fields of imaginary cyclic quartic fields

Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly the Hilbert genus field of the imaginary cyclic quartic fields of the form $\mathbb{Q}(\sqrt{-a\varepsilon_p\sqrt{p}})$.

5-Class towers of cyclic quartic fields arising from quintic reflection

Let $$\zeta _5$$ be a primitive fifth root of unity and $$d\ne 1$$ be a quadratic fundamental discriminant not divisible by $$5$$ . For the $$5$$ -dual cyclic quartic field $${M}={\mathbb {Q}}((\zeta _5-\zeta _5^{-1})\sqrt{d})$$ of the quadratic fields $${k}_1={\mathbb {Q}}(\sqrt{d})$$ and $${k}_2={\mathbb {Q}}(\sqrt{5d})$$ in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group $$\mathrm {G}_5^{(2)}{{M}}=\mathrm {Gal}({M}_5^{(2)}/{M})$$ of the second Hilbert $$5$$ -class field $${M}_5^{(2)}$$ of $${M}$$ are investigated, when the $$5$$ -class group $$\mathrm {Cl}_5(M)$$ is elementary bicyclic of rank two. Usually, the maximal unramified pro- $$5$$ -extension $${M}_5^{(\infty )}$$ of $$M$$ coincides with $${M}_5^{(2)}$$ already. The precise length $$\ell _5{M}$$ of the $$5$$ -class tower of $${M}$$ is determined, when $$\mathrm {G}_5^{(2)}{{M}}$$ is of order less than or equal to $$5^5$$ . Theoretical results are underpinned by the actual computation of all $$83$$ , respectively $$93$$ , cases in the range $$0<d<10^4$$ , respectively $$-2\cdot 10^5<d<0$$ .

On the tame kernels of imaginary cyclic quartic fields with class number one

In this paper, a general architecture has been established for the computation of K2OF, the tame kernel of F, for imaginary cyclic quartic field F=Q(−(D+BD)) with class number one, in particular with large discriminants. As a result, it is proved that K2OF is trivial in the following three cases: B=1, D=2 or B=2, D=13 or B=2, D=29. In the last case, the discriminant of F is 24389. Hence, it can be claimed that the architecture also works for the computation of the tame kernel of a number field with discriminant less than 25000.

A family of cyclic quartic fields with explicit fundamental units

We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.

On the rank of the 2-class group of an extension of degree 8 over $$\mathbb {Q}$$ Q

Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F.

On the p-rank of tame kernel of number fields

In this paper, the relations between p-ranks of the tame kernel and the ideal class group for a general number field are investigated. As a result, nearly all of Browkin's results about quadratic fields are generalized to those for general number fields. In particular, a p-rank formula between the tame kernel and the ideal class group for a totally real number field of odd degree is obtained when p is a Fermat prime. As an example, the case of cyclic quartic fields is considered in more details. More precisely, using the results on cyclic quartic fields, we give some results connecting the p-rank(K2OF) with the p-rank(Cl(OE)), where F is a cyclic quartic field and E is an appropriate subfield of F(ζp).

On certain octic biquartic fields related to a problem of Hasse

In this paper we characterize the monogenity of non-cyclic but abelian octic number fields $$L$$ over the rationals $${{{\varvec{Q}}}},$$ each of which is composed by a linearly disjoint cyclic quartic field of odd prime conductor $$\ell $$ and a quadratic field of prime discriminant $$p^{*}.$$ In the case of each odd conductor $$\ell |p^{*}|,$$ the linear Diophantine equation with unit coefficients in a specified quadratic subfield of $$L$$ is applied to determine the monogenity of the octic field $$L.$$ In the case of each even conductor, among seven octic fields $$L$$ of conductor $$5|2^{*}|,$$ it is shown that the unknown four maximal imaginary subfields $$L$$ of the $$40$$ th cyclotomic field are non-monogenic, in which the non-monogenity of two fields are proved by the evaluation of the absolute norm of a partial factor $${\xi }-{\xi }^{{\rho }}$$ of the different $$\mathfrak {d}({\xi })$$ with a suitable Galois action $${\rho }$$ for any integer $${\xi }$$ in $$L$$ .

HT90 and “simplest” number fields

A standard formula (1.1) leads to a proof of HT90, but requires proving the existence of $\theta$ such that $\alpha\ne0$, so that $\beta=\alpha/\sigma(\alpha)$. We instead impose the condition (M), that taking $\theta=1$ makes $\alpha=0$. Taking $n=3$, we recover Shanks’s simplest cubic fields. The “simplest” number fields of degrees $3$ to $6$, Washington’s cyclic quartic fields, and a certain family of totally real cyclic extensions of $\mathbb{Q} (\cos(\pi/4m))$ all have defining polynomials whose zeroes satisfy (M). Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a $2$-parameter family of octic polynomials with “generic” Galois group ${}_{8}T_{11}$. Imposing an additional algebraic condition on these octics produces a new family of cyclic quartic extensions. This family includes the “simplest” quartic fields and Washington’s cyclic quartic fields as special cases. We obtain more detailed results on our octics when the parameters are algebraic integers in a number field. In particular, we identify certain sets of special units, including exceptional sequences of $3$ units, and give some of their properties.

Congruence formulae modulo powers of 2 for class numbers of cyclic quartic fields

Let K = \( k(\sqrt \theta ) \) be a real cyclic quartic field, k be its quadratic subfield and \( \tilde K = k(\sqrt { - \theta } ) \) be the corresponding imaginary quartic field. Denote the class numbers of K, k and \( \tilde K \) by h K , h k and {417-3} respectively. Here congruences modulo powers of 2 for h − = h K /h K and \( \tilde h^ - = h_{\tilde K} /h_k \) are obtained via studying the p-adic L-functions of the fields.

The proportion of cyclic quartic fields with discriminant divisible by a given prime

An asymptotic formula is given for the number of cyclic quartic fields with discriminant < x and divisible by a given prime.

On the Lagrange resolvents of a dihedral quintic polynomial

The cyclic quartic field generated by the fifth powers of the Lagrange resolvents of a dihedral quintic polynomial f(x) is explicitly determined in terms of a generator for the quadratic subfield of the splitting field of f(x) .

The Index of a Cyclic Quartic Field

Let K be a cyclic quartic field. Let i(K) denote the index of K. It is known that i(K)∈{1, 2, 3, 4, 6, 12}. In Part 1 of this paper we show that i(K) assumes all of these values and we give necessary and sufficient conditions for each to occur. In Part 2 an asymptotic formula is given for the number of cyclic quartic fields with discriminant ≤x and i(K)=i for each i∈{1, 2, 3, 4, 6, 12}.

On the Density of Cyclic Quartic Fields

AbstractAn asymptotic formula is obtained for the number of cyclic quartic fields over Q with discriminant ≤ x.

Congruences for the class numbers of real cyclic sextic number fields

LetK 6 be a real cyclic sextic number field, andK 2,K 3 its quadratic and cubic subfield. Leth(L) denote the ideal class number of fieldL. Seven congruences forh - =h (K 6)/(h(K 2)h(K 3)) are obtained. In particular, when the conductorf 6 ofK 6 is a primep, $$Ch^ - \equiv B\tfrac{{p - 1}}{6}B\tfrac{{5(p - 1)}}{6}(\bmod p)$$ , whereC is an explicitly given constant, andB n is the Bernoulli number. These results on real cyclic sextic fields are an extension of the results on quadratic and cyclic quartic fields.

Q-curves of degree 5 and jacobian surfaces of GL 2 -type

We construct a parametric family {E(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q\(\), and the family {C(s,t,u)} of genus two curves over Q covering E{(+)(s,t,u) whose jacobians are abelian surfaces of GL2-type. We also discuss the modularity for them and the sign change between E{(+)(s,t,u) and its twist E(−)(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces Af attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509.