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If the polynomial has rational roots, for example x 2 − 4x + 4 = (x − 2) 2, or x 2 − 3x + 2 = (x − 2)(x − 1), then the Galois group is trivial; that is, it contains only the identity permutation. In this example, if A = 2 and B = 1 then A − B = 1 is no longer true when A and B are swapped.
() = has at most deg f solutions in /. If p is not prime, then there can potentially be more than deg f ( x ) solutions. Consider for example p=8 and the polynomial f(x)=x 2 -1 , where 1, 3, 5, 7 are all solutions.
The conjecture was disproved in 1966, with a counterexample involving a count of only four different 5th powers summing to another fifth power: 27 5 + 84 5 + 110 5 + 133 5 = 144 5 . Proof by counterexample is a form of constructive proof , in that an object disproving the claim is exhibited.
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in k variables. It was originally described by C. G. Broyden in 1965. [1]Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration.
Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
In modular arithmetic, a number g is a primitive root modulo n if every number a coprime to n is congruent to a power of g modulo n. That is, g is a primitive root modulo n if for every integer a coprime to n, there is some integer k for which g k ≡ a (mod n). Such a value k is called the index or discrete logarithm of a to the base g modulo n.
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.
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