Teorija Galoa

Dijagram rešetke ℚ susednih pozitivnih kvadratnih korena od 2 i 3, njihovih potpolja, i grupa Galoa.

U matematici, teorija Galoa pruža vezu između teorije polja i teorije grupa. Korištenjem teorije Galoa, izvesni problemi u teoriji polja mogu se svesti na teoriju grupa, što je na neki način jednostavnije i lakše razumljivo. Ona je korišćena za rešavanje klasičnih problema, uključujući napor kojim je pokazano da se dva antička problema ne mogu rešiti na način na koji su navedeni (udvostručenje kocke i trisektiranje ugla; treći antički problem, kvadratura kruga, takođe je nerešiv, ali je to pokazano drugim metodima); pokazano je da ne postoji kvintna formula; i pokazano je koji se poligoni mogu konstruisati.

Teorija je nazvana po Evaristu Galoa, koji ju je uveo radi proučavanja korena polinoma i karakterizacije polinomskih jednačina koje su rešive radikalima u smislu svojstava permutacijske grupe njihovih korena - jednačina je rešiva radikalima ako se njeni koreni mogu izraziti formulom koja uključuje samo cele brojeve, n-te korene i četiri osnovne aritmetičke operacije.

Ovu teoriju su popularizovali mnogi matematičari i dalje su je razvili Ričard Dedekind, Leopold Kroneker, Emil Artin i drugi koji su permutacijsku grupu korena tumačili kao grupu automorfizma ekstenzije polja.

Teorija Galoa je bila generalizovana do Galoaovih veza i teorije Grotendika Galoa.

Primene na klasične probleme

Nastanak i razvoj teorije Galoa bio je uzrokovan sledećim pitanjem, koje je bilo jedno od glavnih otvorenih matematičkih pitanja do početka 19. veka:

Da li postoji formula za korene polinomske jednačine petog (ili višeg) stepena u smislu koeficijenata polinoma, koristeći samo uobičajene algebarske operacije (sabiranje, oduzimanje, množenje, deljenje) i primenu radikala (kvadratne korene, kubne korene, etc)?

Abel-Rafinijeva teorema pruža suprotni primer kojim se dokazuje da postoje polinomske jednačine za koje takva formula ne može da postoji. Teorija Galoa daje znatno kompletniji odgovor na ovo pitanje, objašnjavajući zašto je moguće da se reše neke jednačine, uključujući sve one sa stepenom četiri ili manje, u gornjem maniru, i zašto to nije moguće za većinu jednačina stepena pet ili više. Dalje, ona daje konceptualno jasan i lak za transformisanje u algoritam, način da se utvrdi kada se data jednačina višeg stepena može rešiti na taj način.

Teorija Galoa daje jasan uvid u pitanja koja se tiču problema pri konstrukciji lenjirom i šestarom. Ona daje elegantnu karakterizaciju odnosa dužina koji se mogu konstruirati ovom metodom. Koristeći to, postaje relativno lako odgovoriti na klasične probleme geometrije kao su

• Koji se regularni poligoni mogu konstruisati?
• Zašto nije moguće trisektirati svaki ugao pomoću lenjira i šestara?
• Zašto udvostručavanje kocke nije moguće istom metodom?

Istorija

Rana istorija

Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are (up to sign) the elementary symmetric polynomials in the roots. For instance, (x – a)(x – b) = x² – (a + b)x + ab, where 1, a + b and ab are the elementary polynomials of degree 0, 1 and 2 in two variables.

This was first formalized by the 16th-century French mathematician François Viète, in Viète's formulas, for the case of positive real roots. In the opinion of the 18th-century British mathematician Charles Hutton, the expression of coefficients of a polynomial in terms of the roots (not only for positive roots) was first understood by the 17th-century French mathematician Albert Girard; Hutton writes:

...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.

Galois' writings

In 1830 Galois (at the age of 18) submitted to the Paris Academy of Sciences a memoir on his theory of solvability by radicals; Galois' paper was ultimately rejected in 1831 as being too sketchy and for giving a condition in terms of the roots of the equation instead of its coefficients. Galois then died in a duel in 1832, and his paper, "Mémoire sur les conditions de résolubilité des équations par radicaux", remained unpublished until 1846 when it was published by Joseph Liouville accompanied by some of his own explanations. Prior to this publication, Liouville announced Galois' result to the Academy in a speech he gave on 4 July 1843. According to Allan Clark, Galois's characterization "dramatically supersedes the work of Abel and Ruffini."

Aftermath

Galois' theory was notoriously difficult for his contemporaries to understand, especially to the level where they could expand on it. For example, in his 1846 commentary, Liouville completely missed the group-theoretic core of Galois' method. Joseph Alfred Serret who attended some of Liouville's talks, included Galois' theory in his 1866 (third edition) of his textbook Cours d'algèbre supérieure. Serret's pupil, Camille Jordan, had an even better understanding reflected in his 1870 book Traité des substitutions et des équations algébriques. Outside France, Galois' theory remained more obscure for a longer period. In Britain, Cayley failed to grasp its depth and popular British algebra textbooks did not even mention Galois' theory until well after the turn of the century. In Germany, Kronecker's writings focused more on Abel's result. Dedekind wrote little about Galois' theory, but lectured on it at Göttingen in 1858, showing a very good understanding. Eugen Netto's books of the 1880s, based on Jordan's Traité, made Galois theory accessible to a wider German and American audience as did Heinrich Martin Weber's 1895 algebra textbook.

Literatura

• Artin, Emil (1998). Galois Theory. Dover Publications. ISBN 0-486-62342-4.  (Reprinting of second revised edition of 1944, The University of Notre Dame Press).
• Bewersdorff, Jörg (2006). Galois Theory for Beginners: A Historical Perspective. American Mathematical Society. ISBN 0-8218-3817-2. doi:10.1090/stml/035.  .
• Cardano, Gerolamo (1545). Artis Magnæ (PDF) (на језику: Latin). Архивирано из оригинала (PDF) 26. 06. 2008. г. Приступљено 01. 03. 2020. CS1 одржавање: Непрепознат језик (веза)
• Edwards, Harold M. (1984). Galois Theory. Springer-Verlag. ISBN 0-387-90980-X.  (Galois' original paper, with extensive background and commentary.)
• Funkhouser, H. Gray (1930). „A short account of the history of symmetric functions of roots of equations”. American Mathematical Monthly. 37 (7): 357—365. JSTOR 2299273. doi:10.2307/2299273. 
• Hazewinkel Michiel, ур. (2001). „Galois theory”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Jacobson, Nathan (1985). Basic Algebra I (2nd изд.). W. H. Freeman and Company. ISBN 0-7167-1480-9.  (Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)
• Janelidze, G.; Borceux, Francis (2001). Galois Theories. Cambridge University Press. ISBN 978-0-521-80309-0. 
• Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94225-4. 
• Postnikov, M. M. (2004). Foundations of Galois Theory. Dover Publications. ISBN 0-486-43518-0. 
• Rotman, Joseph (1998). Galois Theory (2nd изд.). Springer. ISBN 0-387-98541-7. 
• Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge University Press. ISBN 978-0-521-56280-5. 
• van der Waerden, Bartel Leendert (1931). Moderne Algebra (на језику: German). Berlin: Springer. CS1 одржавање: Непрепознат језик (веза). English translation (of 2nd revised edition): Modern Algebra. New York: Frederick Ungar. 1949.  (Later republished in English by Springer under the title "Algebra".)
• Brian A. Davey and Hilary A. Priestley: Introduction to lattices and Order, Cambridge University Press, 2002.
• Gerhard Gierz, Karl H. Hofmann, Klaus Keimel, Jimmie D. Lawson, Michael W. Mislove, Dana S. Scott: Continuous Lattices and Domains, Cambridge University Press, 2003.
• Marcel Erné, Jürgen Koslowski, Austin Melton, George E. Strecker, A primer on Galois connections, in: Proceedings of the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work, Annals of the New York Academy of Sciences, Vol. 704, 1993, pp. 103–125. (Freely available online in various file formats PS.GZ PS, it presents many examples and results, as well as notes on the different notations and definitions that arose in this area.)
• Mac Lane, Saunders (septembar 1998). Categories for the Working Mathematician (Second изд.). Springer. ISBN 0-387-98403-8. 
• Thomas Scott Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, ISBN 1-85233-905-5.
• Nikolaos Galatos, Peter Jipsen, Tomasz Kowalski, and Hiroakira Ono (2007), Residuated Lattices. An Algebraic Glimpse at Substructural Logics, Elsevier, ISBN 978-0-444-52141-5.
• Garrett Birkhoff: Lattice Theory, Amer. Math. Soc. Coll. Pub., Vol 25, 1940
• Ore, Øystein (1944), „Galois Connexions”, Transactions of the American Mathematical Society, 55: 493—513, doi:10.2307/1990305 
• Grothendieck, A.; et al. (1971). SGA1 Revêtements étales et groupe fondamental, 1960–1961'. Lecture Notes in Mathematics 224. SpringerSphiwe Verlag. 
• Joyal, André; Tierney, Myles (1984). An Extension of the Galois Theory of Grothendieck. Memoirs of the American Mathematical Society. Proquest Info & Learning. ISBN 0-8218-2312-4. 
• Borceux, F. and Janelidze, G., Cambridge University Press (2001). Galois theories, ISBN 0-521-80309-8 (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.)
• Szamuely, T., Galois Groups and Fundamental Groups, Cambridge University Press, 2009.
• Dubuc, E. J and de la Vega, C. S., On the Galois theory of Grothendieck, https://arxiv.org/abs/math/0009145v1
• Gray, Jeremy (2018). A history of abstract algebra: from algebraic equations to modern algebra. Springer Undergraduate Mathematics Series. Cham, Switzerland. ISBN 978-3-319-94773-0. S2CID 125927783. doi:10.1007/978-3-319-94773-0. 
• Kimberling, Clark (1981). „Emmy Noether and Her Influence”. Ур.: Brewer, James W; Smith, Martha K. Emmy Noether: A Tribute to Her Life and Work. Marcel Dekker. стр. 3—61. 
• Kleiner, Israel (2007). Kleiner, Israel, ур. A history of abstract algebra. Boston, Mass.: Birkhäuser. ISBN 978-0-8176-4685-1. doi:10.1007/978-0-8176-4685-1. 
• Monna, A. F. (1975), Dirichlet's principle: A mathematical comedy of errors and its influence on the development of analysis, Oosthoek, Scheltema & Holkema, ISBN 978-9031301751 
• Allenby, R. B. J. T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2 
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1 
• Burris, Stanley N.; Sankappanavar, H. P. (1999) [1981], A Course in Universal Algebra 
• Gilbert, Jimmie; Gilbert, Linda (2005), Elements of Modern Algebra, Thomson Brooks/Cole, ISBN 978-0-534-40264-8 
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third изд.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 
• Sethuraman, B. A. (1996), Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94848-5 
• Whitehead, C. (2002), Guide to Abstract Algebra (2nd изд.), Houndmills: Palgrave, ISBN 978-0-333-79447-0 
• W. Keith Nicholson (2012) Introduction to Abstract Algebra, 4th edition, John Wiley & Sons ISBN 978-1-118-13535-8 .
• John R. Durbin (1992) Modern Algebra : an introduction, John Wiley & Sons
• Charles C. Pinter (1990) [1982] A Book of Abstract Algebra, second edition, from University of Maryland

Spoljašnje veze

• Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed., 1996 Архивирано на сајту Wayback Machine (11. јул 2007)
• An Introduction to Galois Theory
• Fields and Galois Theory - J.S. Milne
• Online textbooks in French, German, Italian and English