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This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is x 5 − x + 1 = 0. Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice.
The Hoffman–Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7, or 57. The Moore graphs are: [3] The complete graphs K n on n > 2 nodes (diameter 1, girth 3, degree n − 1, order n) The odd cycles C 2n+1 (diameter n, girth 2n + 1, degree 2, order 2n + 1). This includes C 5 with diameter 2, girth 5, degree 2 ...
For polynomials in two or more variables, the degree of a term is the sum of the exponents of the variables in the term; the degree (sometimes called the total degree) of the polynomial is again the maximum of the degrees of all terms in the polynomial. For example, the polynomial x 2 y 2 + 3x 3 + 4y has degree 4, the same degree as the term x ...
In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]
Septic equations solvable by radicals have a Galois group which is either the cyclic group of order 7, or the dihedral group of order 14, or a metacyclic group of order 21 or 42. [1] The L(3, 2) Galois group (of order 168) is formed by the permutations of the 7 vertex labels which preserve the 7 "lines" in the Fano plane. [1]
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Abel–Ruffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not ...
This statement, due to Tunnell's theorem (Tunnell 1983), is related to the fact that n is a congruent number if and only if the elliptic curve y 2 = x 3 − n 2 x has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its L-function has a zero at 1). The interest in this statement is that the condition is ...