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More generally, if an equation P(x) = 0 of prime degree p with rational coefficients is solvable in radicals, then one can define an auxiliary equation Q(y) = 0 of degree p – 1, also with rational coefficients, such that each root of P is the sum of p-th roots of the roots of Q.
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 ...
Corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions; [4] also used for denoting Gödel number; [5] for example “āGā” denotes the Gödel number of G. (Typographical note: although the quotes appears as a “pair” in unicode (231C and 231D), they ...
Ramanujan–Skolem's theorem (Diophantine equations) Ramsey's theorem (graph theory, combinatorics) Rank–nullity theorem (linear algebra) Rao–Blackwell theorem ; Rashevsky–Chow theorem (control theory) Rational root theorem (algebra, polynomials) Rationality theorem ; Ratner's theorems (ergodic theory)
The following names are assigned to polynomials according to their degree: [2] [3] [4] Special case – zero (see § Degree of the zero polynomial, below) Degree 0 – non-zero constant [5] Degree 1 – linear; Degree 2 – quadratic; Degree 3 – cubic; Degree 4 – quartic (or, if all terms have even degree, biquadratic) Degree 5 – quintic
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.
An illustration of Newton's method. In numerical analysis, the Newton–Raphson method, also known simply as Newton's method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function.
Noting [16] that it is easier to state for −3 and +5 than it is for +3 or −5, he states [17] the general theorem in the form: If p is a prime of the form 4 n + 1 then p , but if p is of the form 4 n + 3 then − p , is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p .