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	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) (458 words)

	PlanetMath: proof of primitive element theorem
		"proof of primitive element theorem" is owned by mathcam.
		Cross-references: QED, characteristic, fields, fundamental theorem of Galois theory, subgroups, subfields, finite, normal, extension, irreducible polynomial, monic
		This is version 2 of proof of primitive element theorem, born on 2003-01-20, modified 2004-03-22.
	planetmath.org /encyclopedia/ProofOfPrimitiveElementTheorem.html (113 words)

	Primitive Polynomial Computation Theory and Algorithm
		where the field element a is called a generator of the group (or primitive element of the field).
		Any primitive polynomial f(x) of degree n modulo p is the minimal polynomial of a primitive element of the 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)

	Separable extension - Wikipedia, the free encyclopedia
		In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K.
		The effects of inseparability (necessarily for infinite K of characteristic p) can be seen in the primitive element theorem, and for the tensor product of fields.
		Given a finite extension L/K of fields, there is a smallest subfield M of L containing K such that L is a separable extension of M.
	en.wikipedia.org /wiki/Separable_extension (307 words)

	PlanetMath: primitive element theorem
		Note that this implies that every finite separable extension is not only finitely generated, it is generated by a single element.
		For more detail on this theorem and its proof see, for example, Field and Galois Theory, by Patrick Morandi (Springer Graduate Texts in Mathematics 167, 1996).
		This is version 12 of primitive element theorem, born on 2001-10-15, modified 2005-01-22.
	planetmath.org /encyclopedia/PrimitiveElementTheorem.html (245 words)

	List of theorems (Site not responding. Last check: 2007-10-09)
		This is one of the most famous theorems of probability and statistics...
		The Horvitz-Thompson theorem as a unifying perspective for probability sampling: with examples from natural resource sam...
		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.
	hallencyclopedia.com /List_of_theorems (549 words)

	7.3 Index Theorem (Site not responding. Last check: 2007-10-09)
		Theorem 7.2 shows that we can effectively interpret RAM programs.
		The inutitive meaning of Theorem 7.3 is that given any RAM program p of m + n arguments and any set of fixed values
		Observe, that it is possible for this number to be negative (e.g., when each element of the m-tuple x is a 0).
	www.cs.pitt.edu /~daley/cs2110/notes/cs2110w_node30.html (474 words)

	Frege's Logic, Theorem, and Foundations for Arithmetic
		However, for the present purposes, we may note that 0 is defined in terms of the primitive notion ‘the number of Fs’ and a particular complex concept the existence of which is guaranteed in Frege's theory of concepts and second-order logic with identity.
		Theorem 5 now follows from the Lemma on Successors and the fact that successors of natural numbers are natural numbers.
		Frege's Theorem is an elegant derivation of the basic laws of arithmetic which can be carried out independently of the portion of Frege's system which led to inconsistency.
	plato.stanford.edu /entries/frege-logic (15095 words)

	HJM, Vol. 28, No. 1, 2002
		A Theorem is given stating about the harmonicity of 'holomorphic' maps between manifolds of even and odd dimensions (namely almost indefinite (para)-Hermitian and almost (para)- contact (hyperbolic) metric manifolds) in the most general form which gives new results and also covers almost all the known ones.
		The purpose of this paper is to give two theorems of Lorentz-type for complete space-like hypersurfaces with constant mean curvature as an extension of theorems of Akutagawa, Cheng and Nakagawa, and Nishikawa.
		Corrigendum: The primitive element theorem for commutative algebras, pp.
	www.math.uh.edu /~hjm/Vol28-1.html (1316 words)

	Large primitive groups
		The following theorem si proved using the classification of finite simple groups; one can get close by elementary means.
		Symmetry condition imply regularity conditions (e.g., vertex-transitivity is a symmetry condition, which implies that the graph is regular, a regularity condition).
		Using this translation, we shall prove a combinatorial result which implies a nearly optimal upper bound on the order of uniprimitive (primitive but not doubly transitive) permutation groups.
	people.cs.uchicago.edu /~laci/reu03/n2_13/node2.html (278 words)

	Primitive Element Theorem (Site not responding. Last check: 2007-10-09)
		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)

	No Title
		Theorems of Artin and Brauer (characters as linear combinations of induced characters).
		Identification of primitive ideals via rational and locally closed prime ideals (Dixmier-Moeglin equivalence); proof of the equivalence in important examples.
		Classification of primitive ideals by orbits: induced representations, polarizations, the adjoint algebraic group and its action, the Dixmier map, associated tools from noetherian ring theory.
	www.math.ucsb.edu /~mckernan/algebra_courses/algebra_courses.html (1441 words)

	Primitive Element Theorem
		Primitive Element Theorem: If u,v are algebraic over F, a field of characteristic zero, then there's some algebraic w with F(u,v) = F(w).
		Faced by such rising complexity, I see my feeble algebra is inadequate.
		Thus the question, how is the primitive element theorem approached?
	www.usenet.com /newsgroups/sci.math/msg08640.html (93 words)

	Burnside's Theorem (Site not responding. Last check: 2007-10-09)
		Professor Ram Abhyankar pointed out to me that this theorem occurs already in the first edition of Burnside's book (and hence must have a proof not using character theory!) Here is the proof.
		elements of N have order p, and n is a power of p.
		By Theorem 4.4, if a primitive group has more than one minimal normal subgroup, then it has just two, and each is the centraliser of the other, so they are non-abelian and regular.
	www.maths.qmw.ac.uk /~pjc/permgps/burnside.html (241 words)

	SYLLABUS FOR M.A./M.SC. PART II (Site not responding. Last check: 2007-10-09)
		Structure theorem for finitely generated modules, Applications to the theory of a single linear transformation.
		Theorems of Krasnoselski and Rabinowitz, References for this topic are [1] and [4].
		Fourier inversion theorem-Fourier transform is an isomorphism on J plancherel theorem for L(R) distribution on R -support of a distribution-distributions with compact support Convolution for distributions-Tempered distributions-measures-Fourier transform on tempered distribution.
	math.mu.ac.in /syllabus/partII (3320 words)

	Science Chat - Math (Site not responding. Last check: 2007-10-09)
		Suppose M k-subsets are independently drawn from a set P={1,2,....., n} (that is, drawing k elements from P to form a subset and there are M of this.).
		Hi guys, One of the characteristics of a Ring is the absence of an inverse element (regarding multiplication).
		Theorem II 2: The sum theorem for 0-dimensional sets.
	www.science-chat.org /group-4346-14301.html (3544 words)

	Math 649, Lecture Notes (Site not responding. Last check: 2007-10-09)
		Lecture 22 "The Fundamental Theorem of Galois Theory" scheduled for 5 Apr 1996.
		Lecture 25 "The Primitive Element Theorem" scheduled for 15 Apr 1996.
		Lecture 37 "Theorems of Hopkins and Levitzki" scheduled for 29 May 1996.
	darkwing.uoregon.edu /~anderson/math649/notes.html (411 words)

	ADM Seminar
		An effective primitive element theorem for field extensions
		As a consequence, if the number of elements in F is at least m, this result yields an alternate proof of the Primitive Element Theorem.
		We will explore a classical theorem of algebraic geometry known as the Cayley-Bacharach Theorem, and see how a modern version of this theorem may be used to shorten and generalize Hansen's proof.
	www.math.clemson.edu /~gmatthe/adm.html (1919 words)

	Gödel's Incompleteness Theorem
		For the program will eventually generate and check every theorem that can be deduced from the system against every other theorem to insure no theorem is proven to be both true and false.
		Assigning a unique number to each element in a class of objects is known as Gödel numbering.
		These `bits' (elements that can store 0 or 1) are organized in groups of eight known as bytes.
	www.mtnmath.com /whatth/node30.html (682 words)

	Lecture Summary MP473, Number Theory IIIH/IVH
		), the primitive element theorem, norm and trace
		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).
		and f=degree(P), the Kummer-Dedekind theorem describing the prime ideal decomposition of (p) for almost all p, examples: quadratic and cyclotomic fields.
	www.numbertheory.org /courses/MP473/lectures.html (849 words)

	Security Forums Dot Com :: View topic - Primitive element + Public crypto mathematics pointer
		I cannot explain the primitive element (yet, I have to look back in my books on the shelf) but I can explain the math in those others that you mentioned.
		In this case, the conept of a primitive element is different from what I earlier mentioned.
		But there can be a prime smaller than q or a prime larger than q, which may cause the element under test to have a lower order or [higher order but not maximum order], thus preventing the element from being a generator.
	www.security-forums.com /forum/viewtopic.php?p=163745 (2503 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Theorem There is a natural equivalence between the category of algebraic varieties with dominant rational maps and the category of finitely generated field extensions of
		The other follows by interpreting maps of function fields as morphisms defined on open subsets and applying the previous theorem.
		Theorem Every algebraic variety is birationally isomorphic to a hypersurface.
	odin.mdacc.tmc.edu /~krc/agathos/birat.html (345 words)

	Origin of the General Feedback Theorem (GFT)
		In the first attempt [1], a primitive "feedback theorem" showed that the closed-loop gain could be assembled from three transfer functions calculated by breaking the loop at a dependent generator and applying a test signal at the break.
		In the second attempt [2], it was shown that the loop gain could be found by injection of both voltage and current test signals at a "nonideal" point, that is, one that is not at an ideal dependent generator.
		A valuable consequence is that a block diagram built of unidirectional blocks emerges as part of the result, and doesn't have to be guessed at or approximated at the start.
	www.ardem.com /gft_origin.asp (402 words)

	4H Galois Theory 2003-4: revision guide
		Algebraic extensions, minimal polynomials, primitive elements, definition of an algebraic closure of a field, Kronecker's Theorem (both versions), splitting fields, quadratic polynomials and quadratic extension fields.
		Statements of existence of algebraic closures, statement of the Extension Theorem, conjugates and roots of the minimal polynomial of an algebraic element.
		Determining the minimal polynomial of a primitive element in a simple extension.
	www.maths.gla.ac.uk /~ajb/4HGaloisTheory/galoisthy-revision.htm (381 words)

	MTH-3E28: Galois Theory (Site not responding. Last check: 2007-10-09)
		Artin’s Extension Theorem, separability, inseparability, Primitive Element Theorem.
		Normal and Galois extensions, the Fundamental Theorem of Galois Theory, examples of the explicit computation of Galois groups.
		Radical extensions, solvable groups, proof that a polynomial can be solved using radicals if and only if the associated Galois group is a solvable group, radical solution to general quadratic, cubic and quartic equations, explicit examples of polynomials which are not solvable by radicals.
	www.mth.uea.ac.uk /maths/syllabuses/0405/3E2805.html (332 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		(A) Characterize when E/F is Galois in terms of F is the fixed field of the group Aut(E/F) and (for Char zero) E is a splitting field of a polynomial over F. The key for these are Theorems 7 and 9 of section 14.2.
		Two proofs of the fundamental theorem of algebra.
		A finite p-group satifies the converse of Lagrange's theorem.
	www.math.cmu.edu /users/rami/374.S01.review2.html (372 words)

	Algebra I Course Page (Site not responding. Last check: 2007-10-09)
		Groups, subgroups, homomorphisms, normal subgroups, quotient groups, isomorphism theorems, towers of groups, Jordan-Holder series.
		Application of Sylow theorems for classification of groups of small order, counting problems using Cauchy-Frobenius (Burnside's lemma).
		Irreducible polynomials of elements in an algebraic extension.
	www.math.gatech.edu /~saugata/teaching/fall00/6121.html (206 words)

	Abstract Algebra (Math 434), Spring 2003
		State the definition of a prime element in an integral domain.
		State a fundamental theorem that relates degrees of extensions in a tower of three fields.
		Summarize the importance of the Primitive Element Theorem.
	buzzard.ups.edu /courses/2003spring/m434s2003.html (760 words)

	The Fundamental Theorem of Algebra (Site not responding. Last check: 2007-10-09)
		The intermediate value theorem provides a positive square root for every positive real number, and a root to any odd degree polynomial in the reals, as x moves from -
		The first sylow theorem provides a subgroup h whose order is a power of 2, while the index of h is odd.
		Since l/R is finite and separable, apply the primitive element theorem and write l = R(u).
	www.mathreference.com /fld-sep,fta.html (511 words)

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