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See searches and references containing SPLITTING FIELD!SPLITTING FIELD
Field generated by all rupture-fields of a polynomial over a field
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
Splitting_field
Theory in condensed matter physics
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.
Crystal_field_theory
Topics referred to by the same term
Tongue splitting Heegaard splitting Splitting field Splitting principle Splitting theorem Splitting lemma Matrix splitting for the numerical method to
Splitting
List of ligands in coordination compounds topic of Inorganic chemistry
the ligand-field splitting parameter in ligand field theory, or the crystal-field splitting parameter in crystal field theory. The splitting parameter
Spectrochemical_series
Quantum mechanical spectroscopic effect
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
Zero-field_splitting
Algebraic structure
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
Finite_field
Finite dimensional algebra over a field whose central elements are that field
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
Central_simple_algebra
Algebraic structure with addition, multiplication, and division
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_(mathematics)
Equations of degree 5 or higher cannot be solved by radicals
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
Abel–Ruffini_theorem
Mathematical group
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
Galois_group
German mathematician (1882–1935)
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
Emmy_Noether
Type of algebraic field extension
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
Roots of an algebraic element's minimal 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
Conjugate element (field theory)
Conjugate_element_(field_theory)
Algebraic field extension
normal extension and a separable extension. E {\displaystyle E} is a splitting field of a separable polynomial with coefficients in F . {\displaystyle F
Galois_extension
Field arising from a quotient ring by a maximal ideal
ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry
Residue_field
Theorem in differential geometry
In the mathematical field of differential geometry, there are various splitting theorems on when a pseudo-Riemannian manifold can be given as a metric
Splitting_theorem
Correspondence between subfields and subgroups
\mathbb {Q} } , being the splitting field of x 2 + x + 1 {\displaystyle x^{2}+x+1} . Its Galois group over the base field is the quotient group G / H
Fundamental theorem of Galois theory
Fundamental_theorem_of_Galois_theory
Branch of Galois theory in mathematics
{\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
Artin–Schreier_theory
Spectral line splitting in magnetic field
(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
Zeeman_effect
Molecular orbital theory applied to transition metal complexes
Strong-field ligands (such as CN⁻ or CO) produce a large splitting, while weak-field ligands (such as I⁻ or Br⁻) result in a smaller splitting. Tetrahedral
Ligand_field_theory
Construction of a larger algebraic field by "adding elements" to a smaller field
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
Field_extension
Field theory theorem
{\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
Primitive_element_theorem
Generalisation of Fourier transform to any ring
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
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
Energy level splitting due to strong light-matter coupling
Rabi splitting is the separation of a single resonance into two distinct energy or frequency modes when a light field strongly couples to a matter excitation
Rabi_splitting
Algebraic field extension
{\displaystyle S} of K [ x ] {\displaystyle K[x]} , there exists a splitting field of S {\displaystyle S} over K {\displaystyle K} . An algebraic closure
Algebraic_closure
Ring produced from two fields
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
Tensor_product_of_fields
Chemical reaction
Water splitting is the endergonic chemical reaction in which water is broken down into oxygen and hydrogen: 2 H2O → 2 H2 + O2 Efficient and economical
Water_splitting
American actress (born 1946)
Following their 1980 breakup, Field and Reynolds continued to date on and off before splitting permanently in 1982. Field married her second husband, Alan
Sally_Field
Uniform coding for primitive elements of all finite fields
field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define a standard representation of Fpn as a splitting field
Conway polynomial (finite fields)
Conway_polynomial_(finite_fields)
Aspect of algebraic number theory
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
Splitting of prime ideals in Galois extensions
Splitting_of_prime_ideals_in_Galois_extensions
Field extension of the rational numbers by a primitive root of unity
_{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}}
Cyclotomic_field
Decomposition of a compact oriented 3-manifold by dividing it into two handlebodies
In the mathematical field of geometric topology, a Heegaard splitting (Danish: [ˈhe̝ˀˌkɒˀ] ) is a decomposition of a compact oriented 3-manifold that
Heegaard_splitting
Cubic equation unsolvable in real radicals
extension of F by radicals. Then the degree of p is a power of 2, and its splitting field is an iterated quadratic extension of F. Thus for any irreducible polynomial
Casus_irreducibilis
Cubic polynomials defined from a monic polynomial of degree four
group G; that is, the Galois group of the splitting field of P(x). Let m be the degree over k of the splitting field of the resolvent cubic (it can be either
Resolvent_cubic
Describes statistically the splitting of primes in a given Galois extension of Q
Chebotarev, statistically describes the splitting of primes in a given Galois extension K {\displaystyle K} of the field Q {\displaystyle \mathbb {Q} } of rational
Chebotarev_density_theorem
Mathematical function
of Riemann surfaces.) Algebraically, if L {\displaystyle L} is the splitting field of p ( x , y ) {\displaystyle p(x,y)} over C ( x ) {\displaystyle \mathbb
Algebraic_function
Abstract approach to algebraic geometry
finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over
Grothendieck's_Galois_theory
associated linear system defines the d-dimensional embedding of X over a splitting field L. Projective bundle Jacobson (1996), p. 113 Gille & Szamuely (2006)
Severi–Brauer_variety
Topics referred to by the same term
Normal extensions (or quasi-Galois), field extensions, splitting fields for a set of polynomials over the base field Normal family, a pre-compact family
Normal
Algebraic concept
primitive cube root of unity). For a field containing all the roots of a polynomial, see Splitting field. A rupture field of X 2 + 1 {\displaystyle X^{2}+1}
Rupture_field
Every polynomial has a real or complex root
de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p(z). At the end of the 18th century, two new proofs
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
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
Generic_polynomial
Theory in abstract algebra
the rational number field Q, since for three cube roots of 1 complex numbers are required. If one takes L to be the splitting field of X3 − a over Q, where
Kummer_theory
Field theory is the branch of algebra that studies fields
simple extension. Splitting field A field extension generated by the complete factorisation of a polynomial. Normal extension A field extension generated
Glossary_of_field_theory
conjugation in the field K. The set of irreducible characters of G forms an orthogonal basis. Further, if K is a splitting field for G—for instance,
Class_function
Mathematical expression for linear operators
to a Jordan–Chevalley decomposition: If L {\displaystyle L} is the splitting field of the minimal polynomial of x {\displaystyle x} and G {\displaystyle
Jordan–Chevalley decomposition
Jordan–Chevalley_decomposition
strength of this electric field gradient, which is affected by the chemical environment of the nuclei. "Electric quadrupole splitting". Archived from the original
Quadrupole_splitting
Internet error message
content Link rot – URLs ceasing to function List of HTTP status codes Fielding, R; Reschke, J, eds. (June 2014). "404 Not Found". HTTP/1.1 Semantics and
HTTP_404
Method of predicting a chemical complex's absorption spectrum
useful and can be used to approximate the value of 10Dq, the ligand field splitting energy. Tanabe–Sugano diagrams can be used for both high spin and low
Tanabe–Sugano_diagram
Spectral line splitting in electrical field
shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of
Stark_effect
Number constructible via compass and straightedge
sufficient condition for constructibility, one must instead consider the splitting field K = Q ( γ , γ ′ , γ ″ , … ) {\displaystyle K=\mathbb {Q} (\gamma ,\gamma
Constructible_number
Class of internet software vulnerability
HTTP response splitting is a form of web application vulnerability, resulting from the failure of the application or its environment to properly sanitize
HTTP_response_splitting
Polynomial equation of degree two
the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial
Quadratic_equation
Product of pairwise differences
polynomial, the Vandermonde polynomial of its roots is defined over the splitting field; for a non-monic polynomial, with leading coefficient a, one may define
Vandermonde_polynomial
diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields. Split Lie algebras are
Split_Lie_algebra
Nuclear reaction splitting an atom into multiple parts
this nucleus into two alpha particles. The feat was popularly known as "splitting the atom", and would win them the 1951 Nobel Prize in Physics for "Transmutation
Nuclear_fission
Team sport played with sticks and a ball
Field hockey, or simply hockey in Asia, Oceania, Africa and parts of Europe, is a fast-paced team sport in which two teams of eleven players (ten field
Field_hockey
Application layer protocol
Generally, the information of a header field is used by software and not shown to the user. A header field line is formatted as a name-value pair with
HTTP
Opposing approaches to categorisation
principles of lumping and splitting apply to the study of early Christian liturgy. Lumpers, who tend to predominate in this field, try to find a single line
Lumpers_and_splitters
to that 100%. Track and field racers have a variety of options in the ways they can choose to pace their races. Even-splitting is a strategy in which the
Pacing strategies in track and field
Pacing_strategies_in_track_and_field
Topics referred to by the same term
Inseparable polynomial, a polynomial that does not have distinct roots in a splitting field Inseparable (album), by Natalie Cole, 1975 "Inseparable" (song), the
Inseparable
Polynomial coprime with its derivative
discussion.) If L is the field extension K(T1/p), in other words the splitting field of P, then L/K is an example of a purely inseparable field extension. It is
Separable_polynomial
Form of a matrix indicating its eigenvalues and their algebraic multiplicities
assumed to exist over a field extending the base field of the matrix, for instance over the splitting field of p; this field extension does not change
Jordan_normal_form
magnetic field. The Stark effect – splitting because of an external electric field. In physical chemistry: The Jahn–Teller effect – splitting of electronic
Energy_level_splitting
Square matrix constructed from a monic polynomial
{\displaystyle p(x)=(x-\lambda _{1})\cdots (x-\lambda _{n}),} and it has splitting field L = F ( λ 1 , … , λ n ) {\displaystyle L=F(\lambda _{1},\ldots ,\lambda
Companion_matrix
Technique for detecting quantum objects
magnetic field. When the magnetic field frequency is resonant with a spin transition, which means that the frequency matches the energy splitting between
Optically detected magnetic resonance
Optically_detected_magnetic_resonance
Alebraic concept
and let f(X) = Xp − a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose α {\displaystyle \alpha } with
Purely_inseparable_extension
Polynomial equation of degree 3
the field automorphisms that fix K of the smallest extension of K (splitting field). As these automorphisms must permute the roots of the polynomials
Cubic_equation
Elementary particle involved with rest mass
Higgs mass (see plot, right). One way that the Higgs can decay is by splitting into a fermion–antifermion pair. As general rule, the Higgs is more likely
Higgs_boson
Mathematical field obtained by adjunction of nth roots
radical series: a polynomial f over a field K is said to be solvable by radicals if there is a splitting field of f over K contained in a radical extension
Radical_extension
British Army officer (1887–1976)
Field Marshal Bernard Law Montgomery, 1st Viscount Montgomery of Alamein (17 November 1887 – 24 March 1976), was a senior British Army officer who served
Bernard_Montgomery
Branch of mathematics that studies algebraic structures
Field (mathematics) Subfield (mathematics) Multiplicative group Primitive element (field theory) Field extension Algebraic extension Splitting field Algebraically
List of abstract algebra topics
List_of_abstract_algebra_topics
Specific algebraic group
which T {\displaystyle \mathbf {T} } is split, which is called the splitting field of T {\displaystyle \mathbf {T} } . The F {\displaystyle F} -rank of
Algebraic_torus
Field (mathematics) generated by the square root of an integer
{\displaystyle {\mathcal {O}}_{K}} of a quadratic field K {\displaystyle K} . In line with general theory of splitting of prime ideals in Galois extensions, this
Quadratic_field
Polynomial without nontrivial factorization
eventually a field over which P factors into linear factors. This field, unique up to a field isomorphism, is called the splitting field of P. If R is
Irreducible_polynomial
Spectroscopic technique
literature), quadrupole splitting due to atomic-scale electric field gradients; and magnetic splitting due to non-nuclear magnetic fields. Due to the high energy
Mössbauer_spectroscopy
Equations describing classical electromagnetism
displacement field D and the magnetizing field H, while the equations depend only on the free charges Qf and free currents If. This reflects a splitting of the
Maxwell's_equations
American actress (born 1990)
Chicago P.D. (2016–2017), Chicago Justice (2017), The Good Cop (2018), and Splitting Up Together (2018–2019). Her feature film debut was in the independent
Monica_Barbaro
Dynamical Stark effect
molecular dipole transition. In this case, the alternating field has the effect of splitting the two bare transition states into doublets or "dressed states"
Autler–Townes_effect
Group with subnormal series where all factors are abelian
F m {\displaystyle F_{m}} contains a splitting field for f ( x ) {\displaystyle f(x)} The smallest Galois field extension of Q {\displaystyle \mathbb
Solvable_group
Relativistic interaction in quantum physics
orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be
Spin–orbit_interaction
Data item stored in a browser by a website
be used to save information that the user previously entered into form fields, such as names, addresses, passwords, and payment card numbers for subsequent
HTTP_cookie
Stadium in Chicago, Illinois, U.S.
Soldier Field (historically often referred to as Soldiers' Field) is a multi-purpose stadium on the Near South Side of Chicago, Illinois, United States
Soldier_Field
satisfied: L/K is a normal extension and a separable extension, L is a splitting field of a separable polynomial with coefficients in K, |Aut(L/K)| = [L:K]
Glossary_of_number_theory
Election result affecting losing candidate
Academy of Sciences. Other systems exhibit an exit incentive. The vote splitting effect in plurality voting demonstrates this method's strong exit incentive:
Spoiler_effect
Number with an integer power equal to 1
field Q ( exp ( 2 π i / n ) ) . {\displaystyle \mathbb {Q} (\exp(2\pi i/n)).} This field contains all nth roots of unity and is the splitting field
Root_of_unity
Set with associative invertible operation
type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via
Group_(mathematics)
President of Germany from 1925 to 1934
the War Ministry to write the field service regulations on field-engineering and on the use of heavy artillery in field engagements; both were used during
Paul_von_Hindenburg
Non-abelian group of order eight
realized as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field of the polynomial x 8 − 72 x 6 + 180 x 4 − 144 x
Quaternion_group
Biographical film by Antoine Fuqua
personal life, focusing instead on his psychological and artistic journey. Splitting Michael into two films was considered during production, but the production
Michael_(2026_film)
Northern Irish golfer (born 1989)
firm in excess of $25 million plus costs to settle the dispute. After splitting from Horizon in 2013, McIlroy created a new management company, Rory McIlroy
Rory_McIlroy
Social media platform owned by Meta
efforts to give users more control over their posts and accompanying comments field. In July 2016, it announced that users would be able to turn off comments
Island country in the Pacific Ocean
discoveries by notable New Zealanders including Ernest Rutherford for splitting the atom, William Pickering for rocket science, Maurice Wilkins for helping
New_Zealand
Details in the emission spectrum of an atom
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the
Fine_structure
Search engine from Google
Retrieved December 9, 2017. Roberts, Hannah (October 27, 2016). "Google is splitting its search index to target 'stripped down' mobile websites". Business
Google_Search
Electromagnetic effect in physics
when the spin is parallel to the field and − {\displaystyle -} when it is antiparallel. This fact called spin splitting implies that the density of states
Quantum_Hall_effect
Upcoming multi-sport event in Ahmedabad, India
the total to eight sports on the programme. In April 2026, cricket and field hockey were confirmed, and there were proposals made to include up to two
2030_Commonwealth_Games
2026 National Football League championship game
divisional rival, the Los Angeles Rams, in the NFC Championship Game. After splitting two close contests in the regular season, including an overtime matchup
Super_Bowl_LX
Mathematical concept named for Ernst Witt
{\displaystyle k} of characteristic p {\displaystyle p} were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form
Witt_vector
American politician (born 1954)
most toxic, most war-mongering fields from hell, rather than the cheap, clean, green, wholesome and patriotic fields from heaven." Kennedy has advocated
Robert_F._Kennedy_Jr.
SPLITTING FIELD
SPLITTING FIELD
Boy/Male
Biblical
Sitting, or captivity, of the father'.
Surname or Lastname
English (chiefly West Midlands and northern England)
English (chiefly West Midlands and northern England) : topographic name for someone who lived in a house (Middle English hous) in open pasture land (see Field). Reaney draws attention to the form de Felhouse (Staffordshire 1332), and suggests that this may have become Fellows.
Boy/Male
Indian, Sanskrit
Splitting; Opening; Moving Slowly
Boy/Male
Arabic, Muslim, Pashtun
Respectable; Of High Rank; Person Sitting at a High Place
Biblical
sitting, or captivity, of the father
Biblical
overmuch captivity, or sitting
Girl/Female
Biblical
Overmuch captivity, or sitting.
Girl/Female
Indian, Sanskrit
Splitting; Breaking
Boy/Male
Australian, Bengali, Hindu, Indian
King of King; Advancement of King; One who Not Sitting or Resting
Boy/Male
Indian, Sanskrit
Breaking; Splitting
Girl/Female
Biblical
Sitting together.
Biblical
sitting together
Boy/Male
Muslim/Islamic
Person sitting at a high place
Girl/Female
Native American
Butterfly sitting on a flower.
Boy/Male
Muslim
Person sitting at a high place
Boy/Male
Indian
Person sitting at a high place
Biblical
respiration; conversion; taking captive;man sitting in Nob;dweller on the mount, he that predicts;
Female
Native American
Native American Hopi name POLIKWAPTIWA means "butterfly sitting on a flower."
Biblical
the people sitting; or captivity of the people
Boy/Male
Biblical
The people sitting, or captivity of the people.
SPLITTING FIELD
SPLITTING FIELD
Girl/Female
Indian
Boy/Male
Celebrity, Hindu, Indian
Lord of the Universe
Girl/Female
English American Greek Latin
Originally a , Dorothy, or any name ending in -dora. It has become common as a name on its own....
Surname or Lastname
English
English : variant of Claiborne.
Girl/Female
American, British, English
Graceful; Form of Grace
Girl/Female
Arabic, Muslim
Blessing; Loan; Favour
Girl/Female
Arabic, Muslim
Adornment of the World
Boy/Male
Hindu, Indian, Marathi
A Sage
Boy/Male
Hindu, Indian, Telugu
Consciousness; New Year
Boy/Male
Tamil
Feature
SPLITTING FIELD
SPLITTING FIELD
SPLITTING FIELD
SPLITTING FIELD
SPLITTING FIELD
a.
Working smoothly, or without splitting; -- said of timber.
n.
A sitting up of a woman after her confinement, to receive and entertain her friends.
p. pr. & vb. n.
of Split
n.
A cleaving, splitting, or breaking up into parts.
n.
The act of splitting timber by the felt grain.
n.
The actual presence or meeting of any body of men in their seats, clothed with authority to transact business; a session; as, a sitting of the judges of the King's Bench, or of a commission.
v. t.
To cure, by splitting, salting, and smoking.
n.
The act of spitting; expectoration.
n.
Act of spitting out.
n.
The act of paring or splitting leather or skins.
a.
Inclined to spit; spitting much.
n.
The act of cleaving or splitting.
n.
A wooden wedge used in splitting blocks.
p. pr. & vb. n.
of Splint
n.
The act or time of sitting, as to a portrait painter, photographer, etc.
n.
Act of cleaving or splitting.
n.
A tool for splitting wood into shingles; a frow.
a.
Deafening; disagreeably loud or shrill; as, ear-splitting strains.
n.
An iron cleaver or splitting tool; a frow.