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	Galois theory - Wikipedia, the free encyclopedia
		In mathematics, Galois theory is a branch of abstract algebra.
		Galois theory not only provides a beautiful answer to this question, it also explains in detail why it is possible to solve equations of degree four or lower in the above manner, and why their solutions take the form that they do.
		The central idea of Galois theory is to consider those permutations (or rearrangements) of the roots having the property that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted.
	en.wikipedia.org /wiki/Galois_theory (1574 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 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/Galois_field (746 words)

	.:: Galois Field Arithmetic Library ::. (Site not responding. Last check: 2007-10-25)
		Galois field polynomials within the branch are seen as mathematical equivalents of Linear Feed-Back Shift Register (LFSR) and operations upon elements are accomplished via bitwise operations such as xor, and, or logic.
		Galois theory is used to describe and generalize results seen in these fields, for example the AES algorithm can be represented with just a few lines of mathematics using Galois theory and some other related abstract algebra.
		Galois fields are setup by intially defining the size of the field which means how many elements will exist within the field, and also the values those elements will posses.
	www.partow.net /projects/galois (689 words)

	APPENDIX J
		The characteristic of a field is its characteristic as a division algebra.
		In this construction, the field elements (or marks in the language of [Dickson 1900]) are residue classes of the integers J modulo the prime p.
		The prime subfield of a finite field is the submodule of the field generated by unity.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (5929 words)

	[No title]
		The convolution operation is resumed by multiplying and adding the Galois Field the remaining portion of the bits of the second resultant with the remaining portion of the bits of the mask to obtain an encrypted digital information.
		After the Integer Ring operation has completed, the resultant sum is transferred back to the Galois Field as indicated by arrow C, whereupon the remainder of the CRC operation is carried out as indicated by arc D. It will be appreciated that the options for altering the simple CRC process are numerous.
		Since there are no carries in Galois Field arithmetic, the equivalent polynomial q(x) is also dependent on the values of a and k, i.e., it is not a constant.
	www.ece.cmu.edu /~koopman/patents/5398284/5398284.html (3092 words)

	polynomial over gf_p_element (Site not responding. Last check: 2007-10-25)
		The specializations use the representation of the general class polynomial in addition with a reference to an element of the class galois_field characterizing the field over which the polynomial is defined.
		This field must be set explicitely if the polynomial is not the result of an arithmetical operation.
		All arithmetical operations are done in the field over which the polynomials are defined.
	www.math.psu.edu /local_doc/LiDIA/node64.html (754 words)

	[No title]
		Proposition 2.4 establishes a one-to-one correspondence between tensor-closed thick subcategories of B-modules and certain Galois invariant tensor-closed thi* ck subcategories of B L-modules, namely, the image of I. To characterize the image of I, we study the purely inseparable case and the Galois* case separately.
		Suppose K is a field, L is a Galois extension field of K with Galois group G, and R is a graded connected Noetherian graded-commutative K-algebra.
		Suppose B is a finite-dimensional algebra over a field K, and L is a Galois extension field of K. Then the maps I and R of Proposition 2.4 define a one-to-one correspondence between tensor-closed thick subcategories of finitely* * gen- erated B-modules and Galois invariant tensor-closed thick subcategories of fini* *tely generated B L-modules.
	hopf.math.purdue.edu /Hovey-Palmieri/galois.txt (5118 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)

	Sequential Galois multiplication in GF(2.sup.n) with GF(2.sup.m) Galois multiplication gates - Patent 4251875
		In the present invention, a GF(2.sup.n) Galois multiplier is constructed on implemented using a single GF(2.sup.m) Galois multiplier, where m is a positive integral divisor of n greater than 1, i.e., k=n/m as where n=16 and m=2, 4 or 8.
		In the prior art, as in Publication II, it has been shown that Galois field extensions from GF(2.sup.1) to GF(2.sup.2), from GF(2.sup.2) to GF(2.sup.4), and from GF(2.sup.4) to GF(2.sup.8) are extensions of degree 2 and are primitive polynomials of degree 2 to go from the smaller field to the larger field.
		Galois theory shows that m-bit portions of the subfield outputs of the GF(2.sup.m) multiplier can be accumulated simultaneously by sequential Galois adds (bit-wise Exclusive-ORs) to the previous accumulated sum of either subfield outputs or of the subfield outputs multiplied by a specific power of a primitive element as required by the chosen code.
	www.freepatentsonline.com /4251875.html (5809 words)

	12: Field theory and polynomials
		Here a local field is the quotient field of a ring with a unique maximal ideal (such as a power series ring); thus for example the field of Laurent series R((x)) is a local field, as is a p-adic field Q_p.
		Fields of functions of algebraic varieties (essentially the quotient fields of rings F[x1,...,xn]/(P) where P is a multivariable polynomial) are more properly treated in 14: Algebraic Geometry, although these are really just discussions of fields of finite transcendence degree over the ground field.
		Likewise fields of meromorphic functions and local rings of germs of functions are usually treated with their applications to 30: Complex Analysis, 32: Several complex variables, and 58: Analysis on manifolds.
	www.math.niu.edu /~rusin/known-math/index/12-XX.html (1782 words)

	PlanetMath: infinite Galois theory
		be a Galois extension, not necessarily finite dimensional.
		comes equipped with a natural topology, which plays a key role in the statement of the Galois correspondence.
		This is version 2 of infinite Galois theory, born on 2002-05-18, modified 2002-05-18.
	planetmath.org /encyclopedia/KrullTopology.html (220 words)

	CIPO - Canadian Patent Database - Claims - 1226677
		Galois Field GF(2P), being a root of a primitive polynomial 0(X) of degree P, P being equal to M-1, and where the arithmetic operation addition is performed using modulo 2 arithmetic;
		Galois Field GF(2P), being a root of a primitive polynomial Q(X) of degree P, P being equal to M-1 and where the arithmetic operation addition is performed using modulo 2 arithmetic;
		The apparatus of claim 32 wherein said means to multiply a data packet by said Galois field element.alpha.i includes a microprocessor having arithmetic logic capability and capable of accessing memory form which said partial product can be read and readable memory comprising at least 128 bytes said bytes containing said partial products.
	patents1.ic.gc.ca /claims?patent_number=1226677&language=EN_CA (3217 words)

	ABSTRACT ALGEBRA ON LINE: Galois Theory
		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 of characteristic zero, and let E be a radical extension of K. Then there exists an extension F of E that is a normal radical extension of K. Theorem.
		A set that satisfies all the axioms of a field except for commutativity of multiplication is called a division ring or skew field.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	[No title]
		I won't bore you with the details, but it is related to the Galois cohomology of the field in the center of the algebras.
		I was talking about a _Galois extension_ which means that K is a larger field which is formed by adjoining to Q _all_ the roots of this irreducible polynomial.
		The connection with Galois fields is that when the base field is finite, _all_ finite extensions are Galois extensions.
	www.math.niu.edu /~rusin/known-math/98/brauer_gp (1057 words)

	Galois Fields (Site not responding. Last check: 2007-10-25)
		It can be proved that the ring of polynomials over any finite field has at least one irreducible polynomial of every degree.
		as the field and an irreducible polynomial, for what stated in Theorem 16, a new field can be set.
		Moreover, every finite field is isomorphic to some Galois field and any two finite fields with the same numbers of elements are isomorphic.
	www.science.unitn.it /~flego/links/tesi1/node15.html (317 words)

	[No title]
		The indeces and the values of the matrix are field * elements.
		* The array is indexed by the decimal representation of a field * element.
		), where q * is the size of this galois field and m = deg(p).
	www.ida.liu.se /~x99lined/exjobb/benchmarks/GaloisField.java (943 words)

	Open Directory - Science: Math: Algebra: Field Theory (Site not responding. Last check: 2007-10-25)
		Field Arithmetic Archive - This archive stores electronic preprints on the arithmetic of fields, Galois theory, model theory of fields, and related topics.
		Field Theory and Polynomials - Section 12 of the Mathematical Atlas by Dave Rusin.
		Galois Field Package - Allows the use of many Mathematica functions over finite fields without any modification; e.g solving linear equations, inverses, determinants, derivations, resultants.
	dmoz.org /Science/Math/Algebra/Field_Theory (272 words)

	gfdiv (Communications Toolbox)
		See Representing Elements of Galois Fields for an explanation of these formats.
		In all cases, an attempt to divide by the zero element of the field results in a "quotient" of
		m = 2; field = gftuple([-1:p^m-2]',m,p); b = zeros(p^m-1,1); % Numerator is zero since 1 = alpha^0.
	www-rohan.sdsu.edu /doc/matlab/toolbox/comm/gfdiv.html (199 words)

	Galois group and action (Site not responding. Last check: 2007-10-25)
		If it is possible to obtain the full factorization of an integer polynomial over its splitting field, the Galois action of the group of the splitting field can be made entirely explicit.
		Although there exists a polynomial-time algorithm for the factorization of polynomials over number fields, in practice this is the bottleneck for our approach to Galois theory.
		Thus the action on A of such an element of the Galois group is determined by the polynomial (with rational coefficients) expressing the conjugate in a.
	magma.maths.usyd.edu.au /magma/Examples/node63.html (533 words)

	Cryptosystem - Patent 4322577
		The method of encryption and decryption according to the invention is founded on the use of matrices belonging to matrix groups with elements belonging to Galois-field.
		The set of elements differing from 0 in the Galois-fields has the character of a cyclic group implying that each such element in the field can be interpreted as a power of a generating element, also called a primitive element.
		Such an element is a root of an irreducible polynom of degree r with the coefficients belonging to the prime field GF(p) in GF(p.sup.r).
	www.freepatentsonline.com /4322577.html (4636 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		We analyze the structure of the periodic trajectories of the K-system generator of pseudorandom numbers on a rational sublattice which coincides with the Galois field GF[p].
		The period of the trajectories increases as a function of the lattice size p and the dimension of the K-matrix d.
		We emphasize the connection of this approach with the one which is based on primitive matrices over Galois fields.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=535109 (184 words)

	[No title]
		* @param gf the galois field of the coefficients.
		Coefficients are specified as decimal field * elements.
		is not a field * element in the field of the coeffients of
	www.ida.liu.se /~x99lined/exjobb/benchmarks/Polynomial.java (955 words)

	Field and Galois Theory
		This book deals with classical Galois theory, of both finite and infinite extensions, and with transcendental extensions, focusing on finitely generated extensions and connections with algebraic geometry.
		First, it is written to be a textbook for a graduate-level course on Galois theory or field theory.
		To help readers grasp field theory, many concepts are placed in the context of their relationships with other areas of mathematics.
	www.allbookstores.com /book/0387947531 (224 words)

	class LidiaGfq (Site not responding. Last check: 2007-10-25)
		This function assumes the output field Element x has already been constructed, but that it is not already initialized.
		This function assumes the output field base Element x has already been constructed, but that it is not already initialized.
		As Elements are represented by polynom the convert function return the valuation of polynom in characteristic by the Horner Method.
	www.eecis.udel.edu /~linbox/html/LidiaGfq.html (559 words)

	Math 250: Higher Algebra (Fall 2004)
		Since all splitting fields are isomorphic, it makes sense to say that a polynomial f in F[X] has a multiple root, or that two polynomials f,g in F[X] have common roots, in a splitting field of f or fg respectively.
		Call that field K as long as we haven't proved that K=F. Clearly K is in F, and E is the splitting field of g over K; moreover, g is separable, so E/K is normal.
		The intermediate fields are precisely the fields of meromorphic functions for some intermediate lattice L"; since G is abelian, all its subgroups are normal, and thus all these subfields are normal over F'.
	abel.math.harvard.edu /~elkies/M250.04 (12646 words)

	Galois Field Package -- from Mathematica Information Center
		The Galois Field package is an implementation of finite fields in Mathematica.
		The package uses the same operations as Mathematica itself (i.e., +,-,,/,^): this requires one to identify constants and variables to be in a finite field*.
		In this method many original functions of the system become available to finite fields without any modifications, i.e.; solving linear equations, inverses, determinants, derivations, resultants, etc.
	library.wolfram.com /infocenter/MathSource/510 (107 words)

	Nontrivial Galois Module Structure of Cyclotomic Fields, by Marc Conrad and Daniel R. Replogle (Site not responding. Last check: 2007-10-25)
		Abstract: We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property tha t $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-module.
		The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nont rivial Galois module structure.
		However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For $K$ any cyclotomi c field we find an explicit $l$ so that there is a tame degree $l$ extension $L/K$ with nontrivial Galois module structure.
	www.math.uiuc.edu /Algebraic-Number-Theory/0330 (154 words)

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