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Galois theory

In mathematics, Galois theory provides a connection between field theory and group theory. Using Galois theory, certain problems in field theory can be reduced to group theory, in some sense simpler and better understood, it has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated. The subject is named after Évariste Galois, who introduced it for studying the roots of a polynomial and characterizing the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is solvable by radicals if its roots may be expressed by a formula involving only integers, nth roots, the four basic arithmetic operations; the theory has been popularized among mathematicians and developed by Richard Dedekind, Leopold Kronecker, Emil Artin, others who interpreted the permutation group of the roots as the automorphism group of a field extension. Galois theory has been generalized to Grothendieck's Galois theory.

The birth and development of Galois theory was caused by the following question, one of the main open mathematical questions until the beginning of 19th century: Does there exist a formula for the roots of a fifth degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations and application of radicals? The Abel–Ruffini theorem provides a counterexample proving that there are polynomial equations for which such a formula cannot exist. Galois' theory provides a much more complete answer to this question, by explaining why it is possible to solve some equations, including all those of degree four or lower, in the above manner, why it is not possible for most equations of degree five or higher. Further, it gives a conceptually clear, easy to transform into an algorithm, means of telling when some particular equation of higher degree can be solved in that manner. Galois' theory gives a clear insight into questions concerning problems in compass and straightedge construction.

It gives an elegant characterization of the ratios of lengths that can be constructed with this method. Using this, it becomes easy to answer such classical problems of geometry as Which regular polygons are constructible polygons? Why is it not possible to trisect every angle using a compass and a straightedge? Why is doubling the cube not possible with the same method? Galois' theory originated in the study of symmetric functions – the coefficients of a monic polynomial are the elementary symmetric polynomials in the roots. For instance, = x2 – 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 was first understood by the 17th-century French mathematician Albert Girard.

He was the first. In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, negative if and only if there is a pair of distinct complex conjugate roots. See Discriminant:Nature of the roots for details; the cubic was first solved by the 15–16th-century Italian mathematician Scipione del Ferro, who did not however publish his results. This solution was rediscovered independently in 1535 by Niccolò Fontana Tartaglia, who shared it with Gerolamo Cardano, asking him to not publish it. Cardano extended this to numerous other cases, using similar arguments. After the discovery of del Ferro's work, he felt that Tartaglia's method was no longer secret, thus he published his solution in his 1545 Ars Magna, his student Lodovico Ferrari solved the quartic polynomial. In this book, Cardano did not provide a "general formula" for the solution of a cubic equation, as he had neither complex numbers at his disposal, nor the algebraic notation to be able to describe a general cubic equation.

With the benefit of modern notation and complex numbers, the formulae in this book do work in the general case, but Cardano did not know this. It was Rafael Bombelli who managed to understand how to work with complex numbers in order to solve all forms of cubic equation. A further step was the 1770 paper Réflexions sur la résolution algébrique des équations by the French-Italian mathematician Joseph Louis Lagrange, in his method of Lagrange resolvents, where he analyzed Cardano's and Ferrari's solution of cubics and quartics by considering them in terms of permutations of the roots, which yielded an auxiliary polynomial of lower degree, providing a unified understanding of the solutions and laying the groundwork for group theory and Galois' theory. Crucially, however, he did not co

Gmina ƚwidwin

Gmina Świdwin is a rural gmina in Świdwin County, West Pomeranian Voivodeship, in north-western Poland. Its seat is the town of Świdwin; the gmina covers an area of 247.34 square kilometres, as of 2006 its total population is 6,202. Gmina Świdwin contains the villages and settlements of Bedlno, Bełtno, Bierzwnica, Buczyna, Bystrzynka, Cieszyno, Dobrowola, Głuszkowo, Gola Dolna, Gola Górna, Kartlewo, Kawczyno, Klępczewo, Kleśnica, Kłośniki, Kluczkówko, Kowanowo, Krosino, Kunowo, Łąkowo, Lipce, Miłobrzegi, Nowy Przybysław, Osowo, Półchleb, Przymiarki, Rogalinko, Rusinowo, Rycerzewko, Sława, Śliwno, Stary Przybysław, Świdwinek and Ząbrowo. Gmina Świdwin is bordered by the town of Świdwin and by the gminas of Brzeżno, Łobez, Ostrowice, Połczyn-Zdrój, Rąbino, Resko and Sławoborze. Polish official population figures 2006

Mary Lou Finlay

Mary Lou Finlay is a Canadian radio and television journalist, best known for hosting various programs on CBC Radio and CBC Television. Finlay graduated from the University of Ottawa in 1967 with a BA in French literature. For three years she did writing and researching for the Canadian War Museum before her leap to journalism when she began hosting a CBC Ottawa television magazine. In 1975, Finlay moved to Toronto to co-host CBC Television's Take 30, she hosted her own program and Company, in 1976 and 1977 and developed a loyal following. In 1978 she moved to CTV to co-host and produce the award-winning Live It Up!. In 1981 she became co-host with Barbara Frum of CBC Television's nightly current affairs program, The Journal. After the program's first year, Frum remained as sole host and Finlay became a documentary reporter, remaining with the program until 1988. In that year she became host of CBC Radio's current affairs program Sunday Morning, where she remained until the spring of 1994. From 1994 to 1997, she hosted Now CBC Radio's weekly media watchdog program.

Finlay became co-host with Barbara Budd of As It Happens on September 1, 1997, having to cover the death of Diana, Princess of Wales on her first day. She retired following her last appearance on November 30, 2005, a tribute show for Finlay celebrating her years with the CBC. In 2008, she released The As It Happens Files: Radio That May Contain Nuts, a book of reminiscences of her time on the show. Finlay is now a fellow with the Centre for the Study of Democracy at Queen's University in Kingston, Ontario. Martin Goodman Nieman Fellowship at Harvard University. LL. D, Dalhousie University. Meritas Tabaret award, University of Ottawa. Audio interview re: The As It Happens Files

Megan Smolenyak

Megan Smolenyak Smolenyak, born October 9, is an American genealogist and speaker. Smolenyak holds a BSFS in Foreign Service from Georgetown University, an MBA in International Business from George Washington University and an MAS in Information Technology from Johns Hopkins University. Since 1999, she has been a consultant with the U. S. Army's repatriation efforts and has located the families of over a thousand soldiers still unaccounted for from World War I, World War II, Korea and Vietnam; as a genealogist, she is best known for unearthing celebrity roots, conducting forensic research for coroners, police departments, NCIS and the FBI, championing the use of DNA testing to learn about one’s ancestry. She researched Michelle Obama's family tree, researched Annie Moore, the first immigrant through Ellis Island, traced Barack Obama's roots to Moneygall and discovered that Al Sharpton’s great-grandfather had been owned by relatives of Strom Thurmond, she herself is of Irish and Rusyn heritage with roots in Ireland, Slovakia and Ukraine.

Chief Family Historian for Ancestry.com, she founded Unclaimed Persons. Smolenyak has authored six books, she was the winner of a 2009 gold Folio Eddie award, as well as five writing awards from the International Society of Family History Writers and Editors. She has written articles for Ancestry, Family Chronicle, Family Tree Magazine, Irish America, Genealogical Computing, Heritage Quest, NGS NewsMagazine, Everton's Family History Magazine, APG Quarterly. Smolenyak is a Huffington Post contributor. Smolenyak conducted research and wrote the companion book for the U. S. version of Who Do You Think You Are?. Smolenyak has consulted for and appeared on CBS's The Early Show, Good Morning America, the Today Show, Top Chef, CNN, ESPN, BBC Breakfast, African American Lives, PBS's Ancestors, TimeWatch, They Came to America, Who Do You Think You Are?, Faces of America, Finding Your Roots, NuvoTV, NPR, BBC Radio, local television and radio shows, has spoken at the National Genealogical Society, Federation of Genealogical Societies, Who Do You Think You Are Live?, Australasian Federation of Family History Organizations, other historical, military and literary events.

Smolenyak was awarded the 2010 NGS Award of Merit for her work in advancing responsible genealogy to a broad popular audience. She is the recipient of four Telly awards for video production, six magazine writing awards, a former board member of the Association of Professional Genealogists. In Search of Our Ancestors: 101 Inspiring Stories of Serendipity and Connection in Rediscovering Our Family History. Cincinnati, OH: Adams Media Corporation, 2000. ISBN 978-1-58062-317-9 Honoring Our Ancestors: Inspiring Stories of the Quest for Our Roots. Provo, UT: Ancestry.com, 2002. ISBN 978-1-931279-00-0 They Came to America: Finding Your Immigrant Ancestors. San Francisco, CA: Santa Fe Ventures, Inc. 2003. ISBN 978-0-9641403-4-9 Trace Your Roots with DNA: Using Genetic Tests to Explore Your Family Tree.. New York, NY: Rodale, 2004. ISBN 978-1-59486-006-5 Who Do You Think You Are? The Essential Guide to Tracing Your Family History. Viking, 2010. ISBN 0-670-02163-6 Hey, Your Roots Are Showing. Kensington, 2012.

ISBN 978-0-8065-3446-6 Honoring Our Ancestors Megan Smolenyak on Facebook Megan Smolenyak on the Huffington Post Megan Smolenyak on IMDb Official website

Bongnae-dong

Bongnae-dong is a legal dong, or neighbourhood of the Jung-gu district in Seoul, South Korea and governed by its administrative dongs, Sogong-dong and Hoehyeon-dong. Global Logistics System Co. Ltd. the Jungsuck Educational Foundation, the Korea Research Foundation for the 21st Century are in the Hanil Building in Bongnae-dong. Administrative divisions of South Korea "Chronicle of Beopjeong-dong and Haengjeong-dong". Guro-gu Official website. "Mapo Information". The chart of Beopjeong-dong assigned by Haengjeong-dong. Mapo-gu Official website. Archived from the original on 2007-11-05. Jung-gu Official site in English Jung-gu Official site Jung-gu Tour Guide from the Official site Status quo of Jung-gu Resident offices and maps of Jung-gu

Mr Kneebone

Mr Kneebone is the third EP by the Australian rock band Powderfinger. It was released after their first full-length studio album, Parables for Wooden Ears, before the album, Double Allergic, it contains five songs. The EP peaked at #83 on the Australian singles chart, is considered to be "the turning point in Powderfinger's song writing career". Following the lack of success of their first album, Parables for Wooden Ears, despite the moderate success of the EP that preceded it, Powderfinger decided to continue writing and improving their craft and recording music; the group returned to recording with Lachlan "Magoo" Goold, who had produced Transfusion, but chose to record in Metropolis Studios Melbourne where they had recorded Parables. Though the band was signed on a contral with the UK label Polydor Records, the group released the EP independently with the one-off label title Egg the Nest. Powderfinger released the EP at a party at the Roxy in the Valley in Brisbane on 28 July 1995, with local support acts Ammonia and Webster.

At the launch, the group performed songs from all of their releases, played all of the songs from Mr Kneebone. A reviewer from the Australian rock music magazine Time Off commented that "they power into over-drive with the new "I'm Splitting Terry"," the EP's final song, with which the group opened the show; the reviewer said of the launch performance, "Powderfinger embody one of the most exciting futures of Australian rock music. They’re in their element playing loud and live." The cover art was painted by Jolyon Robinson. It was believed to be an image of the group's lead vocalist, Bernard Fanning, but the band notes that this was never the intention while acknowledging the coincidental likeness; as the band are all avid cricket supporters, inside the album sleeve the band thanks, among others, Australian cricketers Mark and Steve Waugh. Bernard Fanning - guitar and vocals John Collins - bass guitar Ian Haug - guitars Darren Middleton - guitars Jon Coghill - drums Art concept - Rachel Liddle, Travers Murr and Powderfinger Graphics layout - Rachel Liddle Paintings - Jolyon Robinson Recorded at Metropolis Studios, MelbourneProduced by Lachlan'Magoo' Goold and Powderfinger Mixed and engineered by Goold and Andy Baldwin Mastered at Studio 301 by Don Bartley All music written and performed by Bernard Fanning, John Collins, Ian Haug, Darren Middleton, Jon Coghill.

Lyrics by Fanning. "Swollen Tongue" – 3:12 "Stitches" – 3:51 "Drongo" – 3:53 "My Urn" – 4:25 "I'm Splittin' Terry" – 3:05