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Topic: Splitting field

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	Splitting field - Wikipedia, the free encyclopedia
		Given an algebraically closed field A containing K, there is a unique splitting field L of P between K and A, generated by the roots of P.
		On the other hand the existence of algebraic closures in general is usually proved by 'passing to the limit' from the splitting field result; which is therefore proved directly to avoid a vicious circle.
		Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extension of K containing K′ that is minimal, in an obvious sense.
	en.wikipedia.org /wiki/Splitting_field (317 words)

	Finite field - Wikipedia, the free encyclopedia
		Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory.
		The multiplicative group of every finite field is cyclic, a special case of a theorem mentioned here in the article about fields.
		Finite fields also find applications in coding theory: many codes are constructed as subspaces of vector spaces over finite fields.
	en.wikipedia.org /wiki/Finite_field (1272 words)

	PlanetMath: finite field (Site not responding. Last check: 2007-10-08)
		This follows from the fact that field extensions obtained from splitting fields are normal extensions.
		Any generator of the multiplicative group of the extension field also algebraically generates the extension field over the base field.
		This is version 7 of finite field, born on 2002-05-03, modified 2006-04-18.
	planetmath.org /encyclopedia/FiniteField.html (540 words)

	PlanetMath: splitting field (Site not responding. Last check: 2007-10-08)
		Theorem: Any polynomial over any field has a splitting field, and any two such splitting fields are isomorphic.
		A splitting field is always a normal extension of the ground field.
		This is version 3 of splitting field, born on 2002-01-05, modified 2002-11-25.
	planetmath.org /encyclopedia/SplittingField.html (101 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Splitting can be conceptualized as"external hashing"or"large-scale hashing", in which the splitting function takes a key and generates a table number, whereas a hashing function takes a key and generates a bucket number within a table.
		One hash function used to split the tables may be based purely upon the data type of the field upon which the data is split, the splitting factor, and the depth of splitting.
		Splitting on the second letter might produce a much more uniform distribution of data, and an implementation could decide to only split on their second letter (by counting to determine the uniformity of a split before performing the split).
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/25962.010412&ELEMENT_SET=DECL (6433 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		Date: 05/16/2004 at 08:58:31 From: Doctor Jacques Subject: Re: Splitting Fields Hi Nina, The computation of Galois groups in general is a very complex question; special, ad hoc, techniques are available for specific cases, and the bi-quadratic equation is one of them.
		In general, the splitting field of an irreducible polynomial of degree n can have degree up to n!; the Galois group is a subgroup of S_n.
		For a general quartic, the degree of the splitting field is therefore a divisor of 24.
	mathforum.org /library/drmath/view/65730.html (823 words)

	Magnetism - Wikipedia, the free encyclopedia
		One tool (often introduced in physics courses) for determining the direction of the velocity vector of a moving charge, the magnetic field, and the force exerted is labeling the index finger "V", the middle finger "B", and the thumb "F".
		Normally, magnetic fields are seen as dipoles, having a "South pole" and a "North pole"; terms dating back to the use of magnets as compasses, interacting with the Earth's magnetic field to indicate North and South on the globe.
		Therefore, when placed in a magnetic field, a magnetic dipole tends to align itself in opposed polarity to that field, thereby canceling the net field strength as much as possible and lowering the energy stored in that field to a minimum.
	en.wikipedia.org /wiki/Magnetic (1935 words)

	SampleOutput.nb
		We set the field K to be the field of rationals Q with a third root of 3 adjoined.
		We declare L2 to be a field obtained by adjoining to the rationals a sixth root of 3.
		Now L is a splitting field extension of K; it is a degree 2 extension given by a square root.
	www.davidson.edu /academic/math/swallow/AlgFieldsWeb/sampleoutputs/SampleOutputMma.htm (1207 words)

	Performance of DBS in a Broad Beam
		The splitting parameters used for all splitting routines were those found to optimize performance in the 6 MV SL25 accelerator with a 10
		Note that with such a large splitting radius, the difference in efficiency with a small change in the splitting radius (eg reducing it by 2 cm so that it exactly encloses the field) is expected to be negligible.
		The directional splitting routines (SBS and DBS) are less efficient in the broad beam simply because of the required increase in splitting field size.
	www.irs.inms.nrc.ca /papers/dbs_paper/node22.html (870 words)

	Galois Implies Separable (Site not responding. Last check: 2007-10-08)
		In summary, the extension f/k is the splitting field for a set of separable polynomials iff f is galois and algebraic.
		As a corollary, any splitting field with base field k is galois, provided the characteristic of k is 0, or k is finite.
		An infinite algebraic extension is galois iff it splits a set of separable polynomials, iff it is the union of finite galois extensions.
	www.mathreference.com /fld-sep,galois.html (508 words)

	Galois Groups (Site not responding. Last check: 2007-10-08)
		An algebraic field is, by definition, a set of elements (numbers) that is closed under the ordinary arithmetical operations of addition, subtraction, multiplication, and division (except for division by zero).
		For example, the set of rational numbers is a field, whereas the integers are not a field, because they are not closed under the operation of division (i.e., the result of dividing one integer by another is not necessarily an integer).
		If it is, then the splitting field E of the polynomial is simply Q itself, and we've already seen that the group of automorphisms of Q that leave Q fixed is nothing but the identity mapping.
	www.mathpages.com /home/kmath290/kmath290.htm (2218 words)

	The GNU Awk User's Guide: Field Splitting Summary (Site not responding. Last check: 2007-10-08)
		Fields are separated by each occurrence of the character.
		Instead, they defer splitting the fields until a field is actually referenced.
		If you really want to split fields on an alphabetic character while ignoring case, use a regexp that will do it for you.
	www.mhatt.aps.anl.gov /dohn/programming/gawk/html/gawk_44.html (316 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-08)
		The splitting field is found by adjoining all of the roots to Q. Thus the splitting field is: Q(cbrt(5), cbrt(5) * zeta_3, cbrt(5) * (zeta_3)^2) This isn't a very pretty way of representing the splitting field, but it is correct.
		F is certainly contained in Q(cbrt(5), zeta_3) since fields are closed under multiplication.
		To see that Q(cbrt(5), zeta_3) is contained in F, we need to write cbrt(5) and zeta_3 as a combination of field operations involving only elements from F. This is quite easy: cbrt(5) = cbrt(5) and zeta_3 = cbrt(5) * zeta_3 / cbrt(5).
	mathforum.org /library/drmath/view/61016.html (353 words)

	Splitting Field (Site not responding. Last check: 2007-10-08)
		The splitting field for s is the smallest extension that splits all the polynomials in s.
		Note that f is the splitting field for s over any intermediate extension between f and k.
		If f does not exist, because k is not embedded in a larger field, or that field is not a complete splitting field for s, we can extend the field, so that it acts as a splitting field for s.
	www.mathreference.com /fld,split.html (270 words)

	The Construction of all Irreducible Modules
		The order of a cyclotomic field must divide the exponent of G. The function constructs all absolutely irreducible representations of G over appropriate extensions or subfields of the field K. The modules returned are non-isomorphic and consist of all distinct modules, subject to the conditions imposed.
		In the case when K is a finite field, the Glasby-Howlett algorithm is used to determine the minimal field over which an irreducible module may be realised.
		Finding irreducible modules over the complex field is straightforward, despite not being able to use the complex field as the field argument.
	www.umich.edu /~gpcc/scs/magma/text976.htm (1485 words)

	transverse terms
		Applied transverse fields are very important tools in the study of MQT in SMMs due, in part, to the high power dependence of the tunnel splitting on the magnitude of the transverse field,, leading to variations of twelve orders of magnitude with transverse fields of a few Tesla.
		In this case, magnetic field distributions generate a tunnel splitting distribution along the molecules of the crystal.
		For example, for a given magnitude of an external transverse field, the tunnel splitting is bigger when the transverse field is applied along the direction of a medium axis than along a hard axis, leading to oscillations of the MQT probability as a function of the angle of orientation of the transverse field.
	www.physics.ucf.edu /~delbarco/html/transverse_terms.html (817 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 8.1
		To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered permutations of the roots that leave the coefficient field fixed.
		Let K be a field, let f(x) be a polynomial in K[x], and let F be a splitting field for f(x) over K. Then Gal(F/K) is called the Galois group of f(x) over K, or the Galois group of the equation f(x) = 0 over K.
		Let K be a field, let f(x) be a polynomial in K[x], and let F be a splitting field for f(x) over K. If f(x) has no repeated roots, then Gal(F/K)
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/81.html (433 words)

	[No title]
		Any simple algebraic field is isomorphic to the field formed by adjoining an indeterminate, x, to the base field, and then factoring out by the two sided ideal generated by the minimum polynomial of the algebraic element over the base field.
		(Taking F as the field, we have F(x) (the field formed by adjoining x, an indeterminate, to F), and F(x)/(pa(x)) is isomorphic to F(a) where pa(x) is the minimum polynomial of a in F, and (pa(x)) is the two sided ideal generated by pa(x) is F(x).
		Since the extension field is merely the base field with the roots appended, the transitivity theorem states that anything in the Galois group moves these roots to each other.
	www.people.virginia.edu /~jba5b/552_midterm.doc (1314 words)

	Chemistry : Chapter 10 : Overview (Site not responding. Last check: 2007-10-08)
		The colors of many solids are due to the crystal field splitting of the d orbitals.
		Those compounds where the electron prefers the higher energy level to pairing in the lower level are called high-spin (or weak field) compounds.
		The energy required for an electron to move from the lower d orbital to the higher one is often in the same range as visible light.
	www.wwnorton.com /chemistry/overview/ch10.htm (1252 words)

	Splitting field: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-08)
		In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by...
		[Click link for more facts about this topic] K is a field extension[For more facts and a topic of this subject, click this link] L of K, EHandler: no quick summary.
		(an algebraic number field (or simply number field) is a finite field extension of the rational numbers q....
	www.absoluteastronomy.com /encyclopedia/s/sp/splitting_field.htm (716 words)

	No Title (Site not responding. Last check: 2007-10-08)
		be fields and L is generated over K by some of the roots of a polynomial f with coefficients in K.
		Prove that M is a splitting field of f over K if and only if M is a splitting field of f over L.
		Prove that in a finite field any element can be written as a sum of at most two squares.
	www.math.gatech.edu /~saugata/teaching/fall00/sol8/sol8.html (336 words)

	Finite Fields
		Unfortunately, the area of field theory is rather large and it would be impossible for us to cover it in detail and still have time to work with the results.
		Furthermore, L is isomorphic to the quotient field K[x]/, where denotes the principal ideal of K[x] generated by f(x).
		We denote the group of all automorphisms of a field L by G(L) and the subgroup of G(L) that fixes all elements of the subfield K of L by G(L/K).
	www-math.cudenver.edu /~wcherowi/courses/finflds.html (3085 words)

	Splitting polynomials and fields: Definitions and motivation (Site not responding. Last check: 2007-10-08)
		Let F be any field, and f be a monic polynomial of degree n in F[X].
		We shall show that every polynomial has a splitting field K, which is unique up to automorphisms.
		In general we can construct a splitting field by adjoining roots one at a time, but then we have to prove that the resulting field does not depend on the order in which we adjoined the roots.
	www.math.harvard.edu /~elkies/M250.04/split.html (183 words)

	The Zeeman Effect
		In the presence of an external magnetic field, these different states will have different energies due to having different orientations of the magnetic dipoles in the external field.
		Including hyperfine structure with the Zeeman effect is more difficult, since the field associated with the proton magnetic dipole moment is weak, and hence it does not take a particularly strong external field to make the Zeeman effect comparable in magnitude to the strength of the hyperfine interactions.
		The approximation of small external field is thus not practical when discussing the Zeeman splitting of hyperfine structure.
	www.pha.jhu.edu /~rt19/hydro/node10.html (564 words)

	Uniqueness of Splitting Fields
		) is the splitting field of p(x) in K (the a_i's are the roots of p(x)).
		We have the splitting field E for p(x) in K and the splitting field E' for p(x) in K'.
		I know there are instances when splitting fields aren't actually equal, but it certainly seems obvious to me that they should be isomorphic (in your example above, it wasn't as obvious, but I think that's partly because it wasn't even obvious that that polynomial even splits in Q[root3+root5]).
	www.physicsforums.com /showthread.php?t=49550&goto=nextoldest (590 words)

	Constant Size - The GNU Awk User's Guide (Site not responding. Last check: 2007-10-08)
		For example, data of this nature arises in the input for old Fortran programs where numbers are run together, or in the output of programs that did not anticipate the use of their output as input for other programs.
		The splitting of an input record into fixed-width fields is specified by assigning a string containing space-separated numbers to the built-in variable
		It is a fatal error to supply a field width that is not a positive number.
	www.gnu.org /software/gawk/manual/html_node/Constant-Size.html (444 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, Section 6.4 (Site not responding. Last check: 2007-10-08)
		An extension field F of K is called a splitting field for f(x) over K if there exist elements r
		Then there exists a splitting field F for f(x) over K, with [F:K] n!.
		Let f(x) be a polynomial over the field K. The splitting field of f(x) over K is unique up to isomorphism.
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/64.html (248 words)

	polylib::splitfield -- the splitting field of a polynomial
		returns a list of two operands: the first one is the splitting field of the polynomial, i.e.
		of the coefficient ring; the second one is a list of all roots of the polynomial in the splitting field, each root followed by its multiplicity.
		The name for the primitive element of the field extension is generated using
	www.mupad.de /doc/25/eng/polylib/splitfld.shtml (256 words)

	AlgFields Functions
		The function returns a description of the declared field, giving its degree over the rationals, names of the algebraic numbers, minimal polynomials of each over the field obtained by adjoining the previous algebraic numbers to the rationals, and complex approximations to the particular roots.
		FDeclareField sets a field K to be the extension field of the rational numbers given by adjoining roots of the polynomials occurring in list.
		FDeclareSplittingExtensionField sets a field L to be the extension field of the field K given by adjoining every root of the polynomial f.
	www.davidson.edu /math/swallow/AlgFieldsWeb/functions.htm (2146 words)

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