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Topic: Primitive element (field theory)

	Primitive element (field theory) - Wikipedia, the free encyclopedia
		In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that
		If the extension L/K admits a primitive element, then L is either a finite extension of K, in case ζ is an algebraic element of L over K, or L is isomorphic to the field of rational functions over K in one indeterminate, if ζ is a transcendental element of L over K.
		The primitive element theorem of field theory answers the question of which finite field extensions have primitive elements.
	en.wikipedia.org /wiki/Primitive_element_(field_theory) (468 words)

	PlanetMath: primitive element theorem (Site not responding. Last check: 2007-10-22)
		Note that this implies that every finite separable extension is not only finitely generated, it is generated by a single element.
		Cross-references: Galois theory, Galois group, minimal polynomials, splitting field, characteristic, corollary, theorem, infinite, algebraic closure, algebraic, clear, roots, polynomial, isomorphic, generate, transcendental, indeterminate, generated by, finitely generated, separable extension, degree, finite, extension, fields
		This is version 12 of primitive element theorem, born on 2001-10-15, modified 2005-01-22.
	planetmath.org /encyclopedia/PrimitiveElementTheorem.html (247 words)

	PlanetMath: proof of primitive element theorem (Site not responding. Last check: 2007-10-22)
		So if we can show that any field extension generated by two elements is also generated by one element, we will be done: simply apply the result to the last two elements
		"proof of primitive element theorem" is owned by archibal.
		This is version 1 of proof of primitive element theorem, born on 2004-03-20.
	planetmath.org /encyclopedia/ProofOfPrimitiveElementTheorem2.html (186 words)

	YourArt.com >> Encyclopedia >> primitive (Site not responding. Last check: 2007-10-22)
		The Amish are a particularly obvious example of this, and the primitive baptist church is another.
		Indigenous peoples and their beliefs and practices are sometimes described as "primitive", a usage that is seen as unhelpful and inaccurate by the vast majority of contemporary anthropologists and similar professionals.
		Primitive communism postulates a pre-agrarian form of communism.
	www.yourart.com /research/encyclopedia.cgi?subject=/primitive (169 words)

	Finite Fields
		K[x] is a principal ideal domain (an ideal is a subring that is closed under multiplication, a principal ideal is an ideal generated by a single element, and a principal ideal domain is a commutative ring with unity all of whose ideals are principal).
		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).
		Now the primitive elements are to be found among the roots of the irreducible polynomials (they cannot be elements of the prime field).
	www-math.cudenver.edu /~wcherowi/courses/finflds.html (3085 words)

	Primitive Polynomial Computation Theory and Algorithm
		The field is exactly the set of all polynomials of degree 0 to n-1 with the two field operations being addition and multiplication of polynomials modulo g(x) and with modulo p integer arithmetic on the polynomial coefficients.
		f(x) is a primitive polynomial iff the field element x generates the cyclic group of non-zero field elements of the finite field.
		A primitive polynomial is the minimal polynomial of a generator, and its roots are conjugates of the generator.
	www.seanerikoconnor.freeservers.com /Mathematics/AbstractAlgebra/PrimitivePolynomials/theory.html (3266 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		This is called finding a primitive element theta of K. If alpha_1,..., alpha_n are the roots of f, we find a primitive element by induction, finding one for Q(theta_i, alpha_{i+1}) where theta_i is a primitive element of Q(alpha_1,..., alpha_i).
		Note that theta_{i-1} is the final theta you want.) The "theorem on the primitive element" says that a primitive element for Q(theta_i, alpha_{i+1}) is of the form theta_i + k * alpha_{i+1} for almost all rational integers k.
		A maximal-degree factor of the resultant should be the minimal polynomial of a primitive element of Q(theta_i, alpha_{i+1}), as long as k is generic enough.
	www.math.niu.edu /~rusin/known-math/95/splitting.fld (619 words)

	Number Theory Glossary
		Fields with a finite number of elements are called Galois fields.
		A Galois Field is a field with finite number of elements.
		A primitive element in a group is an element whose powers exhaust the entire group.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (827 words)

	Primitive element - Wikipedia, the free encyclopedia
		in number theory, a primitive root modulo n
		in field theory, an element that generates a given field extension, see primitive element (field theory)
		in a Hopf algebra, a particular kind of element on which the comultiplication takes a certain simple form.
	en.wikipedia.org /wiki/Primitive_element (123 words)

	The Graduate Research of Johnny Crypto
		A primitive element of a Galois field of size p^n is an element whose powers are all different.
		By definition of a field extension, the field that is being extended is a subfield of the larger field.
		We want to find primitive elements to use for the coefficients of the polynomials that are used to generate the S-Boxes.
	www.artfags.org /~jnycrpto/graduate.htm (495 words)

	Open Questions: Algebraic Number Theory
		Galois theory is a way to "map" extensions of fields to groups and their subgroups in such a way that most of the interesting details about the extension are reflected in details about the groups, and vice versa.
		Class field theory has a reputation for being a very difficult subject, and it is. There is a fair bit of abstract conceptual machinery involved even to explain many of the results, and (of course) much more to prove the results.
		As far as ramification is concerned, the class field K is the maximal abelian extension of k which is unramified except for primes that divide the conductor F. This is an alternative definition of K as the class field of k, so this too is a direct generalization of Hilbert's results.
	www.openquestions.com /oq-ma018.htm (19624 words)

	Galois Theory Glossary
		An element a of a field L is algebraic of degree n over a subfield K if there is an irreducible polynomial f(t) in K[t] of degree n such that f(a) = 0.
		An extension of fields L/K (this notation does not denote any sort of quotient) is a ring homomorphism K --> L. Such a homomorphism has to be injective, so that K is isomorphic to a subfield of L. It is often convenient to identify K with this subfield.
		An element a in a field L is a separably algebraic element over a subfield K if there is a polynomial f(t) in K[t] such that f(a) = 0 and f'(a) is nonzero.
	www.wra1th.plus.com /Galois/gloss.html (892 words)

	Primitive Element Theorem (Site not responding. Last check: 2007-10-22)
		PRIMITIVE ELEMENTS IN FINITE FIELDS WITH ARBITRARY TRACE...
		Primitive element (field theory) article - Primitive element (field theory) math...
		Algorithms for primitive elements of free Lie algebras and Lie superalgebras...
	www.scienceoxygen.com /math/290.html (102 words)

	UNDERSTANDING CONFLICT. VIOLENCE, AND WAR: FOUNDATIONS
		It has led to the development of a field theory of behavior which, I believe, integrates a variety of theoretical and philosophical approaches to war and violence and serves as the phenomenological framework for analyzing whether war is inevitable and what might be done about it.
		The mathematical structure of the field is more precise but loses much of the rich meaning of the field interpretation and narrows communication to the small number familiar with the particular mathematics (linear algebra) involved.
		Dynamic fields entail a continuous extension of energy or potentiality throughout a space or region which is the spring or seat of forces, powers, or influences.
	www.hawaii.edu /powerkills/NOTE10.HTM (6974 words)

	Quantum Field Theory
		Quantum field theory can be developed by adopting the path integral formulism as mentioned earlier, or by combining the field equation with canonical quantization as shown in the next sub-topic.
		A renormalizable theory is one in which the details of a deeper scale are not needed to describe the physics at the present scale, save for a few experimentally measurable parameters (see more in the section about "Renormalizable Theories").
		In the Standard model the scalar field is identified as the Higgs field responsiable for the mass of fermions and gauge bosons.
	universe-review.ca /R15-12-QFT.htm (11980 words)

	Jo on the web.
		Galois Fields GF(q) is a field with q elements, also called a finite field because there are a finite number (q) of elements.
		A Primitive Element of GF(q) is an element ‘a‘such that every field element except zero can be expressed as a power of a.
		The primitive element of such a field would itself be such a polynomial.
	nislab.bu.edu /nislab/projects/bchsync/BCH.html (615 words)

	Document title goes here.
		We'll show how they can be derived from Galois field theory and derive their sharply peaked autocorrelation properties.
		When the Galois field elements are written out as a list of polynomial coefficients, the PN sequence is the last column.
		We don't need to generate the field and do time consuming finite field arithmetic to find a PN sequence, but can use a recursive method instead.
	www.seanerikoconnor.freeservers.com /CommunicationTheory/PseudoNoiseSequences/MechanicalAutocorrelatorBasedOnPseudoNoiseSequences.html (600 words)

	list of theorems - Article and Reference from OnPedia.com
		In some fields, theorem can be considered as a courtesy title, given to major results, although with a content that would not satisfy a mathematician.
		No attempt is made here to comment on that aspect of usage: this is a list of results known as theorems.
		Haboush's theorem (algebraic groups, representation theory, invariant theory)
	www.onpedia.com /encyclopedia/list-of-theorems (172 words)

	The Creative Process in Nature
		It is a tenet of General Systems theory that certain fundamental processes in nature are isomorphic; that if we can understand one of these processes, we can use this knowledge to help us comprehend others of its kind.
		The scientific view of this natural creative process is represented in a 4x3 model of the unified field theory and the creation of matter in the Big Bang, while the occult or mythological creation story, with a similar 4x3 pattern, as found in the astrological tradition, is mapped as an overlay.
		Finally, a numbering system is developed for the elements of counting and a notation system for the operations, relationships, and quantitative concepts of mathematics.
	www.people.cornell.edu /pages/jag8/shaman.html (8441 words)

	12F: Field extensions
		Once upon a time, mathematicians (and others) would spend time on a subject call the "Theory of equations", which was just chock-full of algorithms and the theory of polynomials and their roots.
		Nowadays, this is the subject of ring theory or numerical analysis, but we chose to keep much of that material here since it often involves a consideration of the splitting fields of that polynomial.
		An example from Galois theory: calculating the fixed field K(X)^G, for a certain small G. Example of the fixed field under a subgroup of the Galois group (= Sym(3)).
	www.math.niu.edu /~rusin/known-math/index/12FXX.html (965 words)

	Field Theory
		Prove that the Galois group of a finite extension of finite fields is cyclic.
		Describe the construction of an algebraic closure of a field.
		Show that any finite subgroup of the multiplicative group of a field is cyclic.
	www.math.dartmouth.edu /graduate-students/syllabi/sample-questions/algebra/node5.html (140 words)

	Sophie Huczynska (Site not responding. Last check: 2007-10-22)
		Over the years, number theory has expanded to the study of more complicated structures than the set of natural numbers; it is now a vast subject area, with strong links to other branches of mathematics.
		elements of a finite field which are simultaneously additive and multiplicative generators of the field.
		The result which establishes the existence of such elements is the Primitive Normal Basis Theorem: it states that, for every prime power q and positive integer n, there exists a primitive normal basis of GF(q^n) over GF(q).
	homepages.inf.ed.ac.uk /shuczyns (802 words)

	ABSTRACT ALGEBRA ON LINE: Ideal Theory of Commutative Rings
		Let D be an integral domain with quotient field F, and let I be an ideal of D that is invertible when considered as a fractional ideal.
		Let D be an integral domain with quotient field Q, and let F be a finite extension field of Q. If D* is the set of all elements of F that are integral over D, then D* is a Dedekind domain.
		Let F be a field, and first consider the ring F[x] of polynomials in one indeterminate.
	www.math.niu.edu /~beachy/aaol/commutative.html (2296 words)

	DR Donald Mills (Site not responding. Last check: 2007-10-22)
		My primary interest is in the theory of finite fields and the applications of this theory to communications issues.
		While I was at West Point, I did work regarding the existence of primitive elements in cubic extensions of finite fields where the elements take on the form aC+b, C being a defining element of the cubic extension and a and b belonging to the underlying field.
		The paper corresponding to this is entitled "Primitive roots in cubic extensions of finite fields" (joint with G. McNay), and has recently appeared in the proceedings for the Sixth International Conference on Finite Fields and Applications.
	www.math.siu.edu /mills/default1.htm (2056 words)

	http://www.math.wisc.edu/graduate/guide-qe.htm (Site not responding. Last check: 2007-10-22)
		Galois extensions and the fundamental theorem of Galois theory.
		Theory of Fourier Series; Orthogonal functions; Sturm-Liouville theory and connections with Fourier series; Special Fourier bases (Bessel functions, Legendre polynomials); Fourier transforms (Fourier and Fourier sine and Fourier cosine); Laplace transform and solution of initial-boundary value problems for equations; Evaluation of integrals via complex variables techniques.
		The advanced Model Theory, Recursion Theory, and Set Theory sections correspond, roughly, to the contents of 776, 773, and 771, respectively.
	www.math.wisc.edu /graduate/guide-qe.htm (1829 words)

	SYLLABUS FOR M.A./M.SC. PART II (Site not responding. Last check: 2007-10-22)
		FIELD THEORY: Construction of fields, Algebraic extensions, Transcendence basis transcendence degree, Degree of algebraic extensions,
		Group action and transitive groups, Frobenius kernel and primitive group, Jordan's theorem on sharp multiple transitivity affine and projective geometry, Iwasawa theorem, Witt's theorem and Mathiem groups, Sharp 3-transitive groups, Zassenhaus groups.
		Arithmetic in Quadratic Number fields: Integers, Units Primes and irreducible elements, Failure of unique factorization, (Informal) definition of Ideal class group, Pell's equation and relation to continued fractions.
	math.mu.ac.in /syllabus/partII (3320 words)

	Lecture Summary MP473, Number Theory IIIH/IVH
		The algebraic numbers form a field, the algebraic integers form a ring.
		The field polynomial is a power of the minimal polynomial, properties of norm and trace, splitting fields and normal extensions, K-isomorphisms and K-automorphisms, there are [L : K] K-isomorphims of L, a normal extension L has [L :
		under the isomorphisms of K, discriminant of a field basis, non-vanishing of the discriminant, effect of change of basis on the discriminant, discriminant in terms of conjugates of basis elements, resultant R(f(x),g(x)) of two polynomials, R(f(x),g(x))=0 if and only if f(x) and g(x) have a non-trivial factor in common, Disc(f(x)) the discriminant of f(x).
	www.numbertheory.org /courses/MP473/lectures.html (849 words)

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