The group of pythagorean triples in Number fields

Sunto Generalizzando precedenti definizioni, si introduce il gruppo delle terne pitagoriche nell'anello di interi R di una fissata estensione finita K di Q. Sia S l'anello degli interi di K(i): si dà una descrizione esplicita di tali gruppi in termini delle unità di S, del suo gruppo di classi di ideali e di opportuni ideali primi di S.

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Lavoro eseguito con il contribute del M.P.I.     

Genus fields of real biquadratic fields

Let \(K={\mathbb{Q}}(\sqrt{p},\sqrt{d})\) be a real biquadratic field with prime p≡1 mod 4 and positive integer d≡3 mod 4. In this paper, we give the Hilbert genus field of K explicitly.     

Hilbert genus fields of real biquadratic fields

Let $K=\mathbb{Q}(\sqrt{p},\sqrt{d})$ be a real biquadratic field with p??1 mod 4 or p=2, and d a squarefree positive integer. The Hilbert genus field is described explicitly by Yue (Ramanujan J. 21:17?C25, 2010) in the case that p??1 mod 4 and d??3 mod 4. In this article, we give the Hilbert genus field of K explicitly for the remaining cases. We also consider the function field analogue of this problem.     

Pseudo real closed fields,pseudo p-adically closed fields and NTP2

The main result of this paper is a positive answer to the Conjecture 5.1 of [14] by A. Chernikov, I. Kaplan and P. Simon: If M is a PRC field, then $Th(M)$ is NTP₂ if and only if M is bounded. In the case of PpC fields, we prove that if M is a bounded PpC field, then $Th(M)$ is NTP₂. We also generalize this result to obtain that, if M is a bounded PRC or PpC field with exactly n orders or p-adic valuations respectively, then $Th(M)$ is strong of burden n. This also allows us to explicitly compute the burden of types, and to describe forking.     

On real quadratic function fields

Let q be a power of an odd prime number p and K:=Fq(T) be the rational function field with a fixed indeterminate T. For P a prime of K, let be the maximal real subfield of the Pth-cyclotomic function field and its ring of integers. We prove that there exists infinitely many primes P of even degree such that the cardinal of the ideal class group is divisible by q. We prove also an analogous result for imaginary extensions.     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

Quite complete real closed fields

We prove that any ordered field can be extended to one for which every decreasing sequence of bounded closed intervals, of any length, has a nonempty intersection; equivalently, there are no Dedekind cuts with equal cofinality from both sides. Research supported by the United States-Israel Binational Science Foundation. Publication 757.     

Picard-Vessiot theory for real fields

The existence of a Picard-Vessiot extension for a homogeneous linear differential equation has been established when the differential field over which the equation is defined has an algebraically closed field of constants. In this paper, we prove the existence of a Picard-Vessiot extension for a homogeneous linear differential equation defined over a real differential field K with real closed field of constants. We give an adequate definition of the differential Galois group of a Picard-Vessiot extension of a real differential field with real closed field of constants and we prove a Galois correspondence theorem for such a Picard-Vessiot extension.     

On units of real quadratic fields

Let ∈ = (t + u(d)2/2 be a unit of Q((d)1/2), whose norm = 1. We investigate the properties of the factors of the number u_n which is defined by ∈ⁿ = (t_n + u_n(d)1/2)/2.     

Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

A construction of real closed fields

We introduce a new construction of real closed fields by using an elementary extension of an ordered field with an integer part satisfying . This method can be extend to a finite extension of an ordered field with an integer part satisfying . In general, a field obtained from our construction is either real closed or algebraically closed, so an analogy of Ostrowski's dichotomy holds. Moreover we investigate recursive saturation of an o‐minimal extension of a real closed field by finitely many function symbols.