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The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift . For example consider H = x 2 + x + 2 , in which the coefficient 1 of x is not divisible by any prime, Eisenstein's criterion does not apply to H .
Casus irreducibilis can be generalized to higher degree polynomials as follows. Let p ∈ F[x] be an irreducible polynomial which splits in a formally real extension R of F (i.e., p has only real roots). Assume that p has a root in which is an extension of F by radicals.
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of and is not a divisor of for any k < n. Its roots are all n th primitive roots of unity e 2 i π k n {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers less ...
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. . Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are kn
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
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 ...
Irreducibility (mathematics) In mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over that field. In abstract algebra, irreducible can be an abbreviation for irreducible element of an integral domain; for example an irreducible polynomial.
Let f in F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditions for the irreducible polynomial f to be separable: If E is an extension of F in which f is a product of linear factors then no square of these factors divides f in E[X] (that is f is square-free over E). [6]