In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial...

oxidation state. A higher oxidation state leads to a larger splitting relative to the spherical field. the arrangement of the ligands around the metal ion....

Splitting, railway operation Heegaard splitting Splitting field Splitting principle Splitting theorem Splitting lemma for the numerical method to solve...

the ligand-field splitting parameter in ligand field theory, or the crystal-field splitting parameter in crystal field theory. The splitting parameter...

P form a field of order q, which is equal to F by the minimality of the splitting field. The uniqueness up to isomorphism of splitting fields implies thus...

splitting field over K of pK,α, containing α. If L is any normal extension of K containing α, then by definition it already contains such a splitting...

ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry...

call a field E a splitting field for A over K if A⊗E is isomorphic to a matrix ring over E. Every finite dimensional CSA has a splitting field: indeed...

Zero field splitting (ZFS) describes various interactions of the energy levels of a molecule or ion resulting from the presence of more than one unpaired...

pn elements can be constructed as the splitting field of the polynomial f(x) = xq − x. Such a splitting field is an extension of Fp in which the polynomial...

field extension is the following: Given a polynomial f ( x ) ∈ F [ x ] {\displaystyle f(x)\in F[x]} , let E / F {\displaystyle E/F} be its splitting field...

separable). If L is the field extension K(T 1/p) (the splitting field of P) then L/K is an example of a purely inseparable field extension. In L ⊗ K L {\displaystyle...

extension. For finite extensions, a normal extension is identical to a splitting field. Let L / K {\displaystyle L/K} be an algebraic extension (i.e., L is...

normal extension and a separable extension. E {\displaystyle E} is a splitting field of a separable polynomial with coefficients in F . {\displaystyle F...

same number of preimages. The splitting of primes in extensions that are not Galois may be studied by using a splitting field initially, i.e. a Galois extension...

above construction, one can construct a splitting field of any polynomial from K[X]. This is an extension field L of K in which the given polynomial splits...

In the mathematical field of geometric topology, a Heegaard splitting (Danish: [ˈhe̝ˀˌkɒˀ] ) is a decomposition of a compact oriented 3-manifold that...

simplest case where the Galois group is not abelian. Consider the splitting field K of the irreducible polynomial x 3 − 2 {\displaystyle x^{3}-2} over...

splitting is the chemical reaction in which water is broken down into oxygen and hydrogen: 2 H2O → 2 H2 + O2 Efficient and economical water splitting...

[ˈzeːmɑn]) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch...

of the field automorphisms of the splitting field of the equation that fix the elements of F, where the splitting field is the smallest field containing...

along the same lines that for any subset S of K[x], there exists a splitting field of S over K. An algebraic closure Kalg of K contains a unique separable...

In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric...

{\displaystyle \{1,\alpha ,\ldots ,\alpha ^{n-1},\alpha ^{n}\}} ). If L is a splitting field of f ( X ) {\displaystyle f(X)} containing its n distinct roots α 1...

theorem of Galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups...

the splitting field M of P has Galois group G over L, and such that every extension K/F with Galois group G can be obtained as the splitting field of a...

magnetic field. The Stark effect – splitting because of an external electric field. In physical chemistry: The Jahn–Teller effect – splitting of electronic...

may extend the base field to G F ( q ) {\displaystyle \mathrm {GF} (q)} in order to find a primitive root, i.e. a splitting field for x n − 1 {\displaystyle...

{\displaystyle \beta \in K} , this polynomial is irreducible in K[X], and its splitting field over K is a cyclic extension of K of degree p. This follows since for...

algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field Q {\displaystyle \mathbb {Q} } of rational numbers...