Augmented Solvability of Nth Degree Polynomials

A group is called solvable if its derived series descends to the identity element. Galois discovered that a polynomial is solvable by radicals if and only if its Galois group is solvable. In 1824, Niels Abel published a paper proving the insolvability of a general quintic polynomial. In this paper, we provide two augmented strategies to solve all quintics, and discuss methods for how to make all nth degree polynomials solvable

Algebraic and Integral Closure of a Polynomial Ring in its Power Series Ring

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence arepolynomials. We also show which algebraic numbers appear as radii of convergence for algebraic power series. Additionally, we provide a new characterization of algebraic power series by showing that a convergent power series, α(x), is algebraic over L if and only if α(a) is algebraic over L(a) for every a in the domain of convergence of α(x), where L is a countable subfield of C

Classified by Genre: Rhetorical Genrefication in Cinema

This dissertation argues for a rethinking and expansion of film genre theory. As the variety of media exhibition platforms expands and as discourse about films permeates a greater number of communication media, the use of generic terms has never been more multiform or observable. Fundamental problems in the very conception of film genre have yet to be addressed adequately, and film genre study has carried on despite its untenable theoretical footing. Synthesizing pragmatic genre theory, constructivist film theory, Bourdieusian fan studies, and rhetorical genre studies, the dissertation aims to work through the radical implications of pragmatic genre theory and account for genres role in interpretation, evaluation, and rhetorical framing as part of broader, recurring social activities. This model rejects textualist and realist foundations for film genre; only pragmatic genre use can serve as a foundation for understanding film genres. From this perspective, the concept of genre is reconstructed according to its interpretive and rhetorical functions rather than a priori assumptions about the text or transtextual structures. Genres are not independent structures or relations among texts but performative speech acts about textual relationships and are functions of the rhetorical conditions of their use. This use is not only denotative, but connotative, as well, insofar as certain genre labels evoke aesthetic or moral judgments for certain users. This dissertation proposes the concept of meta-genres, or the sum total of textual and extra-textual attributes plus the evaluative valances a given user associates with a generic label. Meta-genres help guide interpretation and serve as a shorthand for evaluative judgments about certain kinds of films, and are thus central to the kinds of taste politics negotiated through film texts. The rhetorical conditions of genre use can be typified, and this dissertation adapts concepts and methods from the field of rhetorical genre studies to show that the film genre use is most readily observable through its uptake rhetorical genres. These rhetorical genres, in turn, index the social groups and recurring situations that they are called upon to meet. By studying examples like academic writing, popular press reviews, filmmaker interviews, internet message board comments, and digital media recommendation systems, one can identify how specific deployments of generic terms serve as a nexus of text, user, group, and social activities, and can develop a methodology for studying genre as use relative to those dimensions

National Survey of Extension 4-H Professionals\u27 Perceptions of Professional Development Factors

Our study describes factors that influence Extension 4-H professionals\u27 likelihood of participating in a professional development experience. We used a work group–developed survey to measure the level of influence of nine factors. Through convenience sampling, we obtained 558 responses from 4-H professionals representing all regions of the country and multiple job roles. Cost, location, networking, personal needs, and time were strongly influential. Differences related to job role and years of job experience existed. Our major recommendation is that those creating professional development experiences for 4-H professionals consider both the major influential factors involved and factors of adult development, such as the need for self-directed learning