GENERATORS OF FINITE FIELDS WITH PRESCRIBED TRACES

This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.

Combinatorial QFT and the Jacobian Conjecture

The Jacobian Conjecture states that any complex n-dimensional locally invertible polynomial system is globally invertible with polynomial inverse. In 1982, Bass et al. proved an important reduction theorem stating that the conjecture is true for any degree of the polynomial system if it is true in degree three. This degree reduction is obtained with the price of increasing the dimension n. We show in this chapter a result concerning partial elimination of variables, which implies a reduction of the generic case to the quadratic one. The price to pay is the introduction of a supplementary parameter 0≤n′≤n, parameter which represents the dimension of a linear subspace where some particular conditions on the system must hold. We exhibit a proof, in a QFT formulation, using the intermediate field method exposed in Chapter 3.

Algebraic Extensions

Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.

Source mechanisms of seismic events during the 2018 eruption of Sierra Negra Volcano (Galapagos) determined by using polarization properties of complete (near-field and far-field) body waves

<p>Sierra Negra is a basaltic shield volcano in the Galapagos Archipelago (Ecuador) and is the largest of the Galapagos volcanoes. The 2018 eruption was a complex event that included eruptive fissures opening on the northern rim and north-western flank. In this study, we report observations of seismic signals recorded on a temporary dense local network consisting of 14 seismometers and nearby permanent seismic stations, and analyze this data set to retrieve the source mechanisms of moderate pre- and co-eruptive seismic events (body-wave magnitude range of M3.5-5.3). Because of the shallow depths of the seismic events (<2 km) and short source-receiver distances (~1.5-10 km), that are comparable to or shorter than the wavelengths of radiated waves, the effect of near- and intermediate-field terms on dynamic displacements can be significant and hence the far-field assumption may not be well-suited for determining fault-plane solutions. Therefore, we pay special attention on the polarization properties of seismic waves excited at the near-field and intermediate-field ranges, and model and analyze complete displacement wave-fields to determine seismic sources. The source mechanism solutions are also interpreted in light of the volcanic unrest leading to the 2018 eruption, GPS observations, and reported regional centroid moment tensors.</p>