Primitive Elements Research Articles

Formalizing Galois Theory

ABSTRACT We describe a project to formalize Galois theory using the Lean theorem prover, which is part of a larger effort to formalize all of the standard undergraduate mathematics curriculum in Lean. We discuss some of the challenges we faced and the decisions we made in the course of this project. The main theorems we formalized are the primitive element theorem, the fundamental theorem of Galois theory, and the equivalence of several characterizations of finite degree Galois extensions.

On the Existence of Pairs of Primitive and Normal Elements Over Finite Fields

Let $$\mathbb {F}_{q^n}$$ be a finite field with $$q^n$$ elements, and let $$m_1$$ and $$m_2$$ be positive integers. Given polynomials $$f_1(x), f_2(x) \in \mathbb {F}_{q^n}[x]$$ with $$\deg (f_i(x)) \le m_i$$ , for $$i = 1, 2$$ , and such that the rational function $$f_1(x)/f_2(x)$$ satisfies certain conditions which we define, we present a sufficient condition for the existence of a primitive element $$\alpha \in \mathbb {F}_{q^n}$$ , normal over $$\mathbb {F}_q$$ , such that $$f_1(\alpha )/f_2(\alpha )$$ is also primitive.

Invertibility of circulant matrices of arbitrary size

In this paper, we present sufficient conditions to guarantee the invertibility of rational circulant matrices with any given size. These sufficient conditions consist of linear combinations of the entries in the first row with integer coefficients. Our result is general enough to show the invertibility of circulant matrices with any size and arrangement of entries. For example, using these conditions, we show the invertibility of the family of circulant matrices with particular forms of integers generated by a primitive element in . Also, using a combinatorial structure of these sufficient conditions, we show invertibility for circulant 0, 1-matrices.

The Theorem of the Primitive Element

The theorem of the primitive element is one of the basic results of Galois theory. We present a proof, different from the standard one, that is undoubtedly not new, but which in our opinion deserves to be better known. We also show how to find a primitive element in an important special case, a result which does not appear in the standard texts, and which in our opinion deserves to be much better known.

PD53-02 ONCOLOGIC EFFICACY OF PRIMARY RETROPERITONEAL LYMPH NODE DISSECTION FOR PATIENTS WITH PATHOLOGIC N1, N2, AND N3 DISEASE

You have accessJournal of UrologyPenile & Testicular Cancer: Penile & Testicular Cancer II (PD53)1 Sep 2021PD53-02 ONCOLOGIC EFFICACY OF PRIMARY RETROPERITONEAL LYMPH NODE DISSECTION FOR PATIENTS WITH PATHOLOGIC N1, N2, AND N3 DISEASE Sean Kern, Isamu Tachibana, Mohammad Mahmoud, Timothy Masterson, Nabil Adra, Richard Foster, Lawrence Einhorn, and Clint Cary Sean KernSean Kern More articles by this author , Isamu TachibanaIsamu Tachibana More articles by this author , Mohammad MahmoudMohammad Mahmoud More articles by this author , Timothy MastersonTimothy Masterson More articles by this author , Nabil AdraNabil Adra More articles by this author , Richard FosterRichard Foster More articles by this author , Lawrence EinhornLawrence Einhorn More articles by this author , and Clint CaryClint Cary More articles by this author View All Author Informationhttps://doi.org/10.1097/JU.0000000000002080.02AboutPDF ToolsAdd to favoritesDownload CitationsTrack CitationsPermissionsReprints ShareFacebookLinked InTwitterEmail Abstract INTRODUCTION AND OBJECTIVE: Adjuvant chemotherapy (AC) has been advocated after primary retroperitoneal lymph node dissection (RPLND) to reduce the historical 50% risk of relapse in pN2/N3 patients, whereas surveillance is preferred for pN1 patients. We sought to explore the oncologic efficacy of primary RPLND alone in non-seminomatous germ cell tumors in an effort to reduce overtreatment with chemotherapy. METHODS: Patients with pathologic stage II germ cell tumor after primary RPLND between 2007-2017 were identified. Patients were excluded for elevated pre-operative serum tumor markers, receipt of adjuvant chemotherapy, or if pure teratoma or primitive neuro-ectodermal tumor elements were found in the retroperitoneum on pathology. RESULTS: We identified 106 patients to have active germ cell tumor in the retroperitoneum following primary RPLND. Fourteen patients received AC and were excluded from the analysis. There were 56.2% of patients with predominant (>50%) embryonal carcinoma in the orchiectomy specimen and 51% had lymphovascular invasion. Median follow up of the entire cohort was 51.5 months. There were 92 patients treated with RPLND alone of which 39 were pN1, 43 were pN2, and 10 were pN3. Seventy four of the 92 patients have not recurred; their pathologic N stages were 30 pN1, 37 pN2, and 7 pN3. The 5-year recurrence-free survival by pN stage was 85% pN1, 85% pN2, and 63% pN3, respectively (log-rank p-value 0.56, Fig 1). All 18 patients who recurred were treated with first line chemotherapy (usually BEPx3) and remain continuously disease free. Locations of recurrence include biochemical (2), pelvis (2), lung (5), mediastinum (1), suprahilar (1), contralateral/out of field (2), in-field (1 bilateral, 1 right modified, 1 left modified template). Additionally, 2 patients had simultaneous recurrences in the lung/suprahilar region and lung/contralateral retroperitoneum, respectively. No patient died of testicular cancer. CONCLUSIONS: The majority (80.5%) of men with pathologic stage II disease treated with surgery alone in our series never experienced a recurrence. Furthermore, we did not observe a difference in recurrences between pN1 and pN2 patients and thus recommending adjuvant chemotherapy for N2 disease may result in overtreatment in the majority. Source of Funding: None © 2021 by American Urological Association Education and Research, Inc.FiguresReferencesRelatedDetails Volume 206Issue Supplement 3September 2021Page: e916-e916 Advertisement Copyright & Permissions© 2021 by American Urological Association Education and Research, Inc.MetricsAuthor Information Sean Kern More articles by this author Isamu Tachibana More articles by this author Mohammad Mahmoud More articles by this author Timothy Masterson More articles by this author Nabil Adra More articles by this author Richard Foster More articles by this author Lawrence Einhorn More articles by this author Clint Cary More articles by this author Expand All Advertisement Loading ...

On primitive elements of finite fields avoiding affine hyperplanes

Let n≥2 be an integer and let Fq be the finite field with q elements, where q is a prime power. Given Fq-affine hyperplanes A1,…,An of Fqn in general position, we study the existence and distribution of primitive elements of Fqn, avoiding each Ai. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.

An effective version of the primitive element theorem

An effective version of the primitive element theorem

Minimal degrees of algebraic numbers with respect to primitive elements

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree [Formula: see text] Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

Petrogenesis of Neoarchean (2.80–2.75 Ga) Jagannathpur volcanics and the Ghatgaon and Keshargaria dyke swarms, Singhbhum craton, eastern India: Geochemical, Sr[sbnd]Nd isotopic and Sm[sbnd]Nd geochronologic constraints for interaction of enriched-DMM derived magma with metasomatised subcontinental lithospheric mantle

Petrogenesis of Neoarchean dyke swarms and dispersed volcanic successions in the Singhbhum craton can potentially constrain mantle heterogeneity. Their arc-like trace element contents, that have a mismatch with their intracratonic extensional setting, were interpreted to propose widely varied genesis from either an enriched mantle (EM I) or wholly from a metasomatised subcontinental lithospheric mantle (SCLM). In order to test the type of enriched source(s) of magma to the dykes swarms and volcanics, geochemical modeling, Sr and Nd isotopic analyses and geochronology were conducted. The Jagannathpur amygdular volcanics, one such unmetamorphosed volcanic succession, yielded an SmNd isochron age of 2799 ± 67 Ma (2σ) with Ndi = 0.50895 ± 0.00006, MSWD = 0.36 (n = 16) and εNd(T) = −1.12 ± 0.50. As the eruptive age is indistinguishable within error to the Ghatgaon (2762.4 ± 2.0 Ma, 2752.0 ± 0.9 Ma) and the Keshargaria (2800.2 ± 0.7 Ma) dyke swarms, the dykes are interpreted as coeval feeder conduits to the volcanics. Enriched LILE and LREE, Nb/Th and Zr/Th ratios coupled with highly radiogenic present-day Sr isotopic ratios which have mantle-like 87Sr/86Sri ratios (0.7031–0.7055) and negative εNd(T) values (between −1.8 and − 5.6 for the dykes and − 1.3 and − 1.4 for the Jagannathpur volcanics) indicate that the source mantle components underwent enrichment processes likely by pre-2800 Ma subduction. The primitive mantle-normalized trace element patterns for these dykes and volcanics have pronounced negative Nb-Ta-Ti anomalies and their arc-like geochemistry suggest interaction with metasomatised SCLM. Generation of primitive magmas has been modeled by two end member components involving 18% partial melt from enriched-DMM (depleted MORB mantle) [100–70%] and interaction with low degree (5%) partial melt of metasomatised SCLM [0–30%]. These magmas ponded at Moho depths and underwent assimilation-fractional crystallization (AFC) forming the diverse dykes and basaltic andesitic-andesitic volcanics. We link the geophysically imaged Mid Lithospheric Discontinuity to depth of metasomatised SCLM beneath the Singhbhum craton that formed ≥2800 Ma ago.

Primitive elements of the Hopf algebras of tableaux

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.

Melting of metasomatically enriched lithospheric mantle – Constraints from Pan-African monzonites (Damara Orogen, Namibia)

The 560.8 ± 2.4 Ma old Okamutambo Pluton is the easternmost intrusion of the syn-collisional Otjimbingwe Alkaline Complex (OAC; Damara Belt, Namibia) and consists of mafic to intermediate alkaline rocks that belong to the monzodiorite-syenite series. The studied samples are potassic (K2O/Na2O > 1) and have moderately high magnesium (MgO: 5.7–2.6 wt%), nickel (Ni: 66–26 ppm), and chromium (Cr: 223–62 ppm) concentrations. LILE (Ba: 2425–1243 ppm; Sr: 1359–941 ppm) and HFSE (Zr: 447–202 ppm, Nb: 32.8–16.4 ppm, Hf: 4.7–9.4 ppm, Ta: 1.2–2.0 ppm) contents are also high. Strontium and Nd isotope data reveal the existence of two magmatic suites indicating a multi-source origin. Group 1 monzonites-quartz monzonites have moderately evolved Sr and Nd isotopic compositions (initial 87Sr/86Sr: 0.7066 to 0.7073; initial εNd: −3.5 to −5.0) and radiogenic Pb isotope ratios (206Pb/204Pb: 17.65–18.02; 207Pb/204Pb: 15.62–15.67; 208Pb/204Pb: 38.19–38.32). In contrast, group 2 monzodiorites-syenites display more evolved Sr and Nd isotopic compositions (initial 87Sr/86Sr: 0.7088 to 0.7090; initial εNd: −6.7 to −7.1) but similar Pb isotope ratios (206Pb/204Pb: 17.63–17.82; 207Pb/204Pb: 15.64–15.66; 208Pb/204Pb: 38.21–38.37). Differentiation involved AFC processes in group 1 monzonites-quartz monzonites whereas group 2 monzodiorites-syenites were modified by fractional crystallization. Although second-order processes were operative, high total alkali contents, incompatible trace element concentrations in excess of bulk crustal values, and evolved isotopic compositions of the most primitive samples are source-controlled and provide insight into the origin of the parental magmas. With reference to experimental data from the literature, it is inferred that the Okamutambo alkaline rocks represent evolved melts that were generated through melting of enriched lithospheric mantle (phlogopite-lherzolite). The observed negative Nb-Ta and Ti anomalies and positive Pb anomalies in primitive mantle-normalized trace element patterns are in line with a mantle source that contains a recycled crustal component. Isotopic compositions indicate that mantle enrichment is an ancient feature that might be linked to Proterozoic subduction. At 561 Ma, the geodynamic regime in the Damara Orogen was mainly characterized by compression during convergence and continental collision between the Congo and Kalahari cratons which is difficult to reconcile with the generation of the OAC as alkaline magmatism is commonly associated with extensional tectonic regimes. The OAC is, however, associated with a major suture zone that may have been involved in localized transtensional tectonics during oblique flat subduction and thus enabled the generation and ascent of mantle-derived alkaline melts.

Graph Coverings and (Im)primitive Homology: Examples of Low Degree

Given a finite covering of graphs , it is not always the case that is spanned by lifts of primitive elements of . In this article, we study graphs for which this is not the case, and we give here the simplest known nontrivial examples of covers with this property, with covering degree as small as 128. Our first step is focusing our attention on the special class of graph covers where the deck group is a finite p-group. For such covers, there is a representation-theoretic criterion for identifying deck groups for which there exist covers with the property. We present an algorithm for determining if a finite p-group satisfies this criterion that uses only the character table of the group. Finally, we provide a complete census of all finite p-groups of rank and order < 1000 satisfying this criterion, all of which are new examples.

Shadows of Forgotten Ancestors: In search of national representation in the image of folklore

In this case study analysis of the film Shadows of Forgotten Ancestors (1965) directed by Sergei Parajanov (1924–90) I will explore the ways in which traditional dress in this film, as part of a wider imagery of folklore, has been defamiliarized from the ideological canon of social realism. More specifically, I will look at the ways Parajanov’s film, filled with music, dance, colour and ethnographic texture, significantly departed from the traditional representation of non-Russians on the Soviet screen under the Friendship of Peoples policy. Based on a folkloric legend, adapted and published in 1911 by Mykhailo Kotsiubynskyi (1864–1913), Shadows celebrates the ethnographic texture of the Hutsul region by departing significantly from causality and narrative logic and bringing together primitive and modern elements instead. Praised for its authenticity, the film became a turning point in the search for a new site of national expression for Ukrainian filmmakers and more specifically, the role of folklore in its visual presentation. The work of the costume designer Lidiya Bajkova (1905–80) is emblematic in the way it renders authenticity beyond historical, ethnic and material accuracy by seamlessly integrating the costumes into the visual texture of the cinematic image. Her approach demonstrates how motifs and patterns that have previously been delegated to domesticated and melodramatic narratives could conversely become a fundamental substance of the cinematic experience.

«La perfida bestia, si capisce, è la coscienza, colla sua brigata». Tommaso Landolfi nelle regioni dell’indicibile

The present essay aims at examining Landolfi’s poetics at the intersection of the anthropology of Levy-Bruhl, whose concepts of “primitive mentality”, mystical participation of essences and “pre-logic” he embraced, and of the organisation of unconscious psychic processes described by Freud. Through the idea of “primitive mentality”, Landolfi, who was generally reticent toward the ‘talking cure’, seemed to approach a psychoanalysis understood as the practice of regression, which implies leaving aside the position of a conscious subject in order to return to the primitive elements that give structure to the psyche. It is specifically through his choice to materialize the unspeakable that Landolfi placed himself at the intersection of the animistic-anthropological and the psychoanalytic approach: in view of the interdependence between physical and spiritual reality and of the separation of man from the animality in which it is rooted, any determination that is ‘different’, alien to those who discover it (the unspeakable, the impulses, the phobias, the obsessions, the anxieties, etc.) transforms, in Landolfi’s works, into a tangible reality that takes the name and shape of a “beast”.

GENERATORS OF FINITE FIELDS WITH PRESCRIBED TRACES

AbstractThis 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.

A geochemical study of the Crown Formation and Bird Member lavas of the Mesoarchaean Witwatersrand Supergroup, South Africa

Abstract The ca. 2.97 to 2.80 Ga Witwatersrand Supergroup, South Africa, represents the oldest intracontinental sedimentary basin of the Kaapvaal craton. Two volcanic units occur in this supergroup: the widespread Crown Formation lavas in the marine shale-dominated West Rand Group and the more geographically restricted Bird Member lavas, intercalated with fluvial to fluvio-deltaic sandstone and conglomerate of the Central Rand Group. These units remain poorly studied as they are rarely exposed and generally deeply weathered when cropping out. We report whole-rock major and trace elements, Hf and Nd-isotope whole-rock analyses of the lavas from core samples drilled in the south of the Witwatersrand basin and underground samples from the Evander Goldfield in the northeast. In the studied areas, both the Crown Formation and Bird Member are composed of two units of lava separated by sandstone. Whereas all the Crown Formation samples show a similar geochemical composition, the upper and lower volcanic units of the Bird Member present clear differences. However, the primitive mantle-normalized incompatible trace element concentrations of all Crown Formation and Bird Member samples show variously enriched patterns and marked negative Nb and Ta anomalies relative to Th and La. Despite the convergent geodynamic setting of the Witwatersrand Supergroup suggested by the literature, the Crown Formation and Bird Member are probably not related to subduction-related magmatism but more to decompression melting. Overall, the combined trace element and Sm-Nd isotopic data indicate melts from slightly to moderately depleted sources that were variably contaminated with crustal material. Greater contamination, followed by differentiation in different magma chambers, can explain the difference between the two signatures of the Bird Member. Finally, despite previous proposals for stratigraphically correlating the Witwatersrand Supergroup to the Mozaan Group of the Pongola Supergroup, their volcanic units are overall geochemically distinct.

An Argument from Normativity for Primitive Emotional Phenomenology

Uriah Kriegel has attempted to describe the varieties of consciousness, that is, the primitive elements that constitute the phenomenal realm. Perceptual, imaginative, algedonic, cognitive, entertaining, and conative are the types of phenomenology acknowledged by him. This list, though right, is incomplete. My main claim is that for it to be complete it should include sui generis emotional phenomenology. To motivate that thesis, I will highlight the characteristic normativity of emotional phenomenology and contrast it with the characteristic normativity of Kriegel’s phenomenal elements. I will conceive, by means of a mental experiment in the form of a phenomenal contrast, a felt normative clash between emotional phenomenology and Kriegel’s primitive phenomenal states. The idea behind it is that the felt normative clash is possible because emotional phenomenology and Kriegel’s phenomenal elements are metaphysically distinct.

Existence of primitive 2-normal elements in finite fields

An element α∈Fqn is normal over Fq if B={α,αq,αq2,⋯,αqn−1} forms a basis of Fqn as a vector space over Fq. It is well known that α∈Fqn is normal over Fq if and only if gα(x)=αxn−1+αqxn−2+⋯+αqn−2x+αqn−1 and xn−1 are relatively prime over Fqn, that is, the degree of their greatest common divisor in Fqn[x] is 0. Using this equivalence, the notion of k-normal elements was introduced in Huczynska et al. (2013): an element α∈Fqn is k-normal over Fq if the greatest common divisor of the polynomials gα[x] and xn−1 in Fqn[x] has degree k; so an element which is normal in the usual sense is 0-normal.Huczynska et al. made the question about the pairs (n,k) for which there exist primitive k-normal elements in Fqn over Fq and they got a partial result for the case k=1, and later Reis and Thomson (2018) completed this case. The Primitive Normal Basis Theorem solves the case k=0. In this paper, we solve completely the case k=2 using estimates for Gauss sum and the use of the computer, we also obtain a new condition for the existence of k-normal elements in Fqn.

Ethnomathematics: Primary School Mathematics Contents on Lio Traditional House at Waka Village in Ende NTT

This study aims to explore elementary school math concepts in traditional houses in Waka village, Ende, NTT. This research is a qualitative research with an ethnographic approach. The process of collecting data in this study was carried out by interview, observation and documentation. The results showed that the Waka traditional house was related to elementary school mathematics. Mathematical concepts found in traditional houses in Waga Village are dominated by the concepts of primitive geometric elements (such as points, lines, line segments, angles) and flat and space forms.

Plane Curves Which are Quantum Homogeneous Spaces

Let mathcal {C} be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, mathcal {C} is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring mathcal {O}(mathcal {C}) of mathcal {C} as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.