Galois Theory: Abel-Ruffini Theorem, Abhyankar's Conjecture, Absolute Galois Group, Artin-Schreier Theory, Biquadratic Field, Embeddi

Overview

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Galois Theory, Galois Group, Abel-ruffini Theorem, Fermat's Last Theorem, Wiles' Proof of Fermat's Last Theorem, P-Adic Hodge Theory, Splitting of Prime Ideals in Galois Extensions, Quintic Equation, Inverse Galois Problem, Frobenius Endomorphism, Fundamental Theorem of Galois Theory, Gaussian Period, Absolute Galois Group, ...

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Overview

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Chapters: Galois Theory, Galois Group, Abel-ruffini Theorem, Fermat's Last Theorem, Wiles' Proof of Fermat's Last Theorem, P-Adic Hodge Theory, Splitting of Prime Ideals in Galois Extensions, Quintic Equation, Inverse Galois Problem, Frobenius Endomorphism, Fundamental Theorem of Galois Theory, Gaussian Period, Absolute Galois Group, Galois Cohomology, Field Arithmetic, Septic Equation, Generic Polynomial, Hasse-arf Theorem, Galois Module, Grothendieck-katz P-Curvature Conjecture, Embedding Problem, Galois Extension, Abhyankar's Conjecture, Haran's Diamond Theorem, Artin-schreier Theory, Sextic Equation, Biquadratic Field. Excerpt: In algebra , the Abel Ruffini theorem (also known as Abel's impossibility theorem ) states that there is no general solution in radicals to polynomial equations of degree five or higher. Interpretation The content of this theorem is frequently misunderstood. It does not assert that higher-degree polynomial equations are unsolvable. In fact, the opposite is true: every polynomial equation in one unknown, with real or complex coefficients, has at least one complex number as solution; this is the fundamental theorem of algebra . Although the solutions cannot always be expressed exactly with radicals, they can be computed to any desired degree of accuracy using numerical methods such as the Newton Raphson method or Laguerre method , and in this way they are no different from solutions to polynomial equations of the second, third, or fourth degrees. The theorem only concerns the form that such a solution must take. The theorem says that not all solutions of higher-degree equations can be obtained by starting with the equation's coefficients and rational constants, and repeatedly forming sums, differences, products, quotients, and radicals ( n -th roots, for some in...

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Product Details

  • ISBN-13: 9781155668666
  • Publisher: General Books LLC
  • Publication date: 5/6/2010
  • Pages: 40
  • Product dimensions: 7.44 (w) x 9.69 (h) x 0.08 (d)

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