 Page "Associative algebra" ¶ 43
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Every and polynomial Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic. Every polynomial P in x corresponds to a function, ƒ ( x ) Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial. * Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ). Every delta operator ' has a unique sequence of " basic polynomials ", a polynomial sequence defined by three conditions: Every real polynomial of odd degree has at least one real number as a root. * Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial. * Every Jacobi-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q * Every Laguerre-like polynomial sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality is, and has Q = * Every Hermite-like polynomial sequence can have its domain shifted and / or scaled so that its interval of orthogonality is, and has Q Every polynomial function is a rational function with. * Every irreducible polynomial over k has distinct roots. * Every polynomial over k is separable. Every field and every polynomial ring over a field ( in arbitrarily many variables ) is a reduced ring. In mathematics, an integer-valued polynomial ( also known as a numerical polynomial ) P ( t ) is a polynomial whose value P ( n ) is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. * Every irreducible polynomial in K which has a root in L factors into linear factors in L.

Every and ring ** Every unital ring other than the trivial ring contains a maximal ideal. Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra. In Norse mythology, Draupnir ( Old Norse " the dripper ") is a gold ring possessed by the god Odin with the ability to multiply itself: Every ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original. Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable. Every division ring is therefore a division algebra over its center. Every objective has a different size ring, so for every objective another condenser setting has to be chosen. Every match must be assigned a rule keeper known as a referee, who is the final arbitrator ( In multi-man lucha libre matches, two referees are used, one inside the ring and one outside ). * In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime. * Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal. Every Boolean ring R satisfies xx Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings. Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )). * Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. * Every localization of a commutative Noetherian ring is Noetherian. Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner. Every prostaglandin contains 20 carbon atoms, including a 5-carbon ring.

Every and R * Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) = Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >. Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism. Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >. * Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm. Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module. Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra. * Every left ideal I in R is finitely generated, i. e. there exist elements a < sub > 1 </ sub >, ..., a < sub > n </ sub > in I such that I = Ra < sub > 1 </ sub > + ... + Ra < sub > n </ sub >. * Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion. Every year since 1982, the W. C. Handy Music Festival is held in the Florence / Sheffield / Muscle Shoals area, featuring blues, jazz, country, gospel, rock music and R & B. Every adult citizen of this small settlement signed the small petition ; E. K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and R. H. Valentine. Every smooth submanifold of R < sup > n </ sup > has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R < sup > n </ sup >.

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