About the Book
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 56. Chapters: Abel's binomial theorem, Abel Ruffini theorem, Addition theorem, Alperin Brauer Gorenstein theorem, Amitsur Levitzki theorem, Ax Grothendieck theorem, Beck's monadicity theorem, Binomial inverse theorem, Birkhoff's representation theorem, Boolean prime ideal theorem, Brauer Cartan Hua theorem, Cartan Dieudonne theorem, Chevalley Warning theorem, Classification of finite simple groups, Cohn's irreducibility criterion, Complex conjugate root theorem, Cramer's rule, Crystallographic restriction theorem, Descartes' rule of signs, Factor theorem, Frobenius determinant theorem, Fundamental lemma (Langlands program), Fundamental theorem of algebra, Fundamental theorem of cyclic groups, Fundamental theorem of Galois theory, Generic flatness, Gershgorin circle theorem, Going up and going down, Hahn embedding theorem, Haran's diamond theorem, Harish-Chandra isomorphism, Haynsworth inertia additivity formula, Higman's embedding theorem, Hilbert's irreducibility theorem, Hilbert Burch theorem, Hudde's rules, Krull's principal ideal theorem, Krull Akizuki theorem, Liouville's theorem (differential algebra), Mason Stothers theorem, Mitchell's embedding theorem, Mori Nagata theorem, Multinomial theorem, Nielsen Schreier theorem, Niven's theorem, Norm residue isomorphism theorem, Polynomial remainder theorem, Rational root theorem, Segal conjecture, Sinkhorn's theorem, Skolem Noether theorem, Specht's theorem, Stone's representation theorem for Boolean algebras, Strassmann's theorem, Subring test, Sylow theorems, Sylvester's determinant theorem, Sylvester's law of inertia, Weil conjecture on Tamagawa numbers, Witt's theorem. Excerpt: The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with zero imaginary part. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two statements can be proven through the use of successive polynomial division. In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals (or some other equivalent formulation of completeness), which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. Peter Rothe, in his book Arithmetica Philosophica (published in 1608), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algebre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds unless the equation is incomplete, by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation x = 4x 3, although incomplete, has four solutions (counting multiplicities): 1 (twice), 1 + i, and 1 i . As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients c
