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...

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...

isomorphism of splitting fields implies thus that all fields of order q {\displaystyle q} are isomorphic. Also, if a field F {\displaystyle F} has a field of order...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

_{n}} , so Q ( ζ n ) {\displaystyle \mathbb {Q} (\zeta _{n})} is the splitting field of x n − 1 {\displaystyle x^{n}-1} (or of Φ n {\displaystyle \Phi _{n}}...

{\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...

{\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...

(Dutch: [ˈzeːmɑn]) is the splitting of a spectral line into several components in the presence of a static magnetic field. It is caused by the interaction...

associated linear system defines the d-dimensional embedding of X over a splitting field L. Projective bundle Jacobson (1996), p. 113 Gille & Szamuely (2006)...

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...

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...

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

simple extension. Splitting field A field extension generated by the complete factorisation of a polynomial. Normal extension A field extension generated...

Following their 1980 breakup, Field and Reynolds continued to date on and off before splitting permanently in 1982. Field married her second husband, Alan...

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...