© Hindawi Publishing Corp. A RELATIVE INTEGRAL BASIS OVER Q ( √ −3) FOR THE NORMAL CLOSURE OF A PURE CUBIC FIELD

Let K be a pure cubic field. Let L be the normal closure of 
 K . A relative integral basis (RIB) for L over ℚ ( − 3 ) is given. This RIB simplifies and 
completes the one given by Haghighi (1986).


Introduction.
Let K be the pure cubic field Q(d 1/3 ), where d is a cube-free integer, and let L be the normal closure of K so that Q ⊂ K ⊂ L, [L : K] = 2, and The first is that in certain cases the RIB makes use of an element of norm 3 in a pure cubic field, a quantity which is not easy to determine, see [2,Theorem 5.1].The second problem is that the RIB is not completely general, see [2,Theorem 5.3].In this note, we give a simple and completely general RIB for L/k.

Preliminary remarks.
As d is a cube-free integer, we can define integers a and b by and if a 2 ≡ b 2 (mod 9), an integral basis is These integral bases are due to Dedekind [1].From (2.2) and (2.3), we deduce that the discriminant d(K) of K is given by where (2.5) The relative discriminant d(L/k) of L/k is given by see [1].We note that if α, β ∈ O L are such that then {1,α,β} is a RIB for L/k.

RIB for L/k.
We show that {1,α,β} is a RIB for L/k, where α and β are given in Table 3.1.
An easy calculation making use of (2.2), (2.3), (2.4), and (2.5) shows that so that (2.7) holds in view of (2.6).Clearly, α ∈ L and β ∈ L. We now show that α ∈ Ω and β ∈ Ω so that α ∈ O L and β ∈ O L , proving that {1,α,β} is a RIB for L/k.Clearly, α ∈ Ω in Cases (i) and (iii), and β ∈ Ω in Cases (ii) and (iv), see (2.3) for the latter.In the remaining cases, it suffices to give a monic polynomial f α (x) ∈ Z[x] of which α is a root in Cases (ii) and (iv), and a monic polynomial of which β is a root in Cases (i) and (iii).

Case (i).
Here, Case (ii).Here, Case (iii).We have and we define m ∈ Z by In this case, Case (iv).We have Here, This completes the proof that {1,α,β} is a RIB for L/k.
We conclude with four examples.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: and [k : Q] = 2.The ring of all algebraic integers is denoted byΩ.The rings of integers of K, k, L are O K = K ∩ Ω, O k = k ∩ Ω, O L = L ∩ Ω,respectively.As O k is a principal ideal domain, L/k possesses a relative integral basis (RIB) [3, Corollary 3, page 401].Haghighi [2, Theorems 5.1, 5.3, 5.6] has given a RIB for L/k.However, Haghighi's RIB for L/k contains two difficulties.