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Topic: Frobenius automorphism

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	NationMaster - Encyclopedia: Automorphism (Site not responding. Last check: 2007-11-01)
		In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.
		An automorphism of a differentiable manifold M is a diffeomorphism from M to itself.
		In Riemannian geometry an automorphism is a self-isometry.
	www.nationmaster.com /encyclopedia/Automorphism (2983 words)

	Automorphism - Wikipedia, the free encyclopedia
		In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges.
		An automorphism of a differentiable manifold M is a diffeomorphism from M to itself.
		In Riemannian geometry an automorphism is a self-isometry.
	en.wikipedia.org /wiki/Automorphism (887 words)

	Frobenius endomorphism - Wikipedia, the free encyclopedia
		In commutative algebra and field theory, which are branches of mathematics, the Frobenius endomorphism is a special endomorphism of rings with prime characteristic, a class importantly including fields.
		The iterates of the Frobenius map are also used in defining the Frobenius closure and tight closure of an ideal.
		In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K.
	en.wikipedia.org /wiki/Frobenius_automorphism (1044 words)

	PlanetMath: Frobenius homomorphism
		If it is surjective then it is an automorphism, and is called the Frobenius automorphism.
		Note: This morphism is sometimes also called the “small Frobenius” to distinguish it from the map
		This map is then also referred to as the “big Frobenius” or the “power Frobenius map”.
	www.planetmath.org /encyclopedia/FrobeniusAutomorphism3.html (116 words)

	Science Fair Projects - Frobenius automorphism
		In mathematics, the Frobenius automorphism is an automorphism induced by a prime power mapping defined for various extensions of fields.
		L/F is always a Galois extension, and the corresponding Galois group is cyclic with the Frobenius automorphism as a generator.
		Given an unramified finite extension L/K of local fields, there is a concept of Frobenius automorphism which induces the Frobenius automorphism in the corresponding extension of residue fields.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Frobenius_automorphism (751 words)

	Frobenius automorphism - Definition, explanation
		L/F is always a Galois extension, and the corresponding Galois group is cyclic with the Frobenius automorphism as a generator.
		Given an unramified finite extension L/K of local fields, there is a concept of Frobenius automorphism which induces the Frobenius automorphism in the corresponding extension of residue fields.
		In algebraic number theory, Frobenius elements are defined for extensions L/K of global fields that are finite Galois extensions for prime ideals Φ of L that are unramified in L/K.
	www.calsky.com /lexikon/en/txt/f/fr/frobenius_automorphism.php (646 words)

	[No title]
		When Q is unramified: G_Q isomorphic to Gal_k l; definition of the Frobenius symbol of Q in L:K. Concrete characterization of the Frobenius symbol.
		20000303: Setup for the Frobenius density theorem (abelian case): extension L:K of number fields, commutative Galois group G. First form of the theorem: for any subgroup H of G, the density of primes with Frobenius symbol in H is #H/#G. Proof.
		Typical application (already proved by Frobenius): density of primes with a given factorization type in L is the density of permutations with that cycle type in the Galois closure of L. What the existence theorem says.
	cr.yp.to /2000-515/inclass.html (3147 words)

	Finite Fields and Polynomials over them
		Prove that any automorphism of a field is the identity map on its simple subfield.
		Apply the Frobenius automorphism to its alleged irreducible factor.
		The operation is the composition, the unit is the identity map, the inverse is the inverse map.
	www.podval.org /~sds/finfield.html (666 words)

	PlanetMath: Frobenius homomorphism
		is a field homomorphism, called the Frobenius homomorphism, or simply the Frobenius map on
		Cross-references: morphism, automorphism, surjective, field homomorphism, map, characteristic, field
		This is version 8 of Frobenius homomorphism, born on 2002-02-18, modified 2006-10-04.
	planetmath.org /encyclopedia/FrobeniusAutomorphism3.html (116 words)

	Automorphisms of Local Rings and Fields (Site not responding. Last check: 2007-11-01)
		The automorphisms of a local ring or field are determined by their images on the inertial element generating the inertia ring and on the uniformizing element.
		Every automorphism of L is determined by the images of pi and a.
		Since the inertia ring is isomorphic to a Galois extension of the p-adic ring by a, its automorphisms are induced by the powers of the Frobenius automorphism of GF(p, f), mapping a to a^p.
	www.sci.kuniv.edu.kw /magma/text506.html (285 words)

	GAP Manual: 18.11. FrobeniusAutomorphism (Site not responding. Last check: 2007-11-01)
		The Frobenius automorphism f of a finite field F of characteristic p is the function that takes each element z of F to its p-th power.
		Each automorphism of F is a power of the Frobenius automorphism.
		Thus the Frobenius automorphism is a generator for the Galois group of F (and an appropriate power of it is a generator of the Galois group of F over a subfield S) (see GaloisGroup).
	www.math.uiuc.edu /Software/GAP-Manual/FrobeniusAutomorphism.html (94 words)

	Ch14-AutosOverFiniteFIelds.html
		The pair of worksheets on automorphisms ooked at automorphisms of finite extensions of the rationals.
		For a finite field an automorphism is defined by the image of a generating element and extended by linearity.
		Notice that in all cases where the map is not an automorphism, the problem shows up when i is the degree of the polynomial f(x) used to generate the field.
	www.adeptscience.com /products/mathsim/maple/powertools/abstractalgebra/html/Ch14-AutosOverFiniteFIelds.html (1366 words)

	Ch14-AutosOverFiniteFIelds.html
		The pair of worksheets on automorphisms ooked at automorphisms of finite extensions of the rationals.
		For a finite field an automorphism is defined by the image of a generating element and extended by linearity.
		Notice that in all cases where the map is not an automorphism, the problem shows up when i is the degree of the polynomial f(x) used to generate the field.
	adept.maplesoft.com /powertools/abstractalgebra/html/Ch14-AutosOverFiniteFIelds.html (1366 words)

	Math Forum - Ask Dr. Math
		Date: 04/14/2004 at 08:52:58 From: Sebastian Subject: Abstract algebra, Frobenius automorphism Hi Dr. Math.
		Date: 04/30/2004 at 10:39:29 From: Doctor Nitrogen Subject: Re: Abstract algebra, frobenius automorphism Hi, Sebastian: You can use this argument for your proof of (a) and (b) above.
		Any element from the field Z_p is left fixed by the Frobenius automorphism s_p, as, by Fermat's Little Theorem, a_p is congruent to a modulo p.
	mathforum.org /library/drmath/view/65377.html (556 words)

	[No title]
		(I'm assuming you have no topological or other restrictions on which automorphisms you allow.) For example, partition the elements of (a countable subset of) a basis into {e_1, e_2}, {e_3, e_4, e_5},...; then let f be the automorphism which cyclically permutes the elements in each set.
		For such an F, all automorphisms of an n-dimensional vector space V over F are periodic.
		To see this, choose a basis for V and describe an automorphism with respect to this basis by a matrix.
	www.math.niu.edu /~rusin/known-math/98/aut_v_periodic (920 words)

	Geometry Seminar Lecture Notes 1
		An automorphism of a field is a bijection of the field onto itself which is both an additive and multiplicative homomorphism (i.e., preserves both addition and multiplication).
		For a finite field GF(q) of characteristic p, the map f(x) = x
		Theorem: The automorphism group of GF(q) of characteristic p is cyclic and generated by the Frobenius automorphism.
	www-math.cudenver.edu /~wcherowi/geom/gsln1.html (1161 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Title : Mathematical Sciences: Geometric Stability Theory Abstract : 9400894 Hrushovski The investigator would like to study the model theory of the Frobenius automorphism.
		On the one hand, he formulated a conjecture analogous to the Lang-Weil estimates, but more general, that would determine the first order theory of the Frobenius.
		On the other hand, together with several coworkers, he is in the process of carrying out the fundamental model theoretic analysis of the theory of difference fields in question, along the lines of Shelah and Zilber.
	www.cs.utexas.edu /users/yguan/NSFAbstracts/Abstracts/MPS/DMS.MPS.a9400894.txt (288 words)

	SVIBOR - Papers - project code: 1-01-261
		Here it is shown that these are (up to isomorphism) the only (78,22,6) designs with such an automorphism group.
		The full classification of symmetric block designs (78,22,6) withan automorphism group isomorphic to the semidirect product of theelementary abelian group of order 8 with the Frobenius group oforder 21 is carried out.
		Since the non-existence of(81,16,3)-designs with involutory automorphism fixing 17 pointshas already been proved, it follows that any involution of(81,16,3)-design must fix just 9 points.
	www.mzos.hr /svibor/1/01/261/rad_e.htm (1623 words)

	DC MetaData for: Cyclic Codes and the Frobenius Automorphism (Site not responding. Last check: 2007-11-01)
		Abstract:Using the theory of cyclic codes some results connected with the Frobenius automorphism are proved.
		For which $GF(q^n)$ exists for any $k = 0,..., n$ exactly one subspace $C$ with dim $C = k$ and which is invariant under the Frobenius automorphism.
		Furthermore, an additional short proof of the formula for the number of normal bases is given.
	www.mathematik.tu-bs.de /preprints/shadow/199728_shadow.html (88 words)

	Commutative Algebra Seminar, Fall 2005 (Site not responding. Last check: 2007-11-01)
		Wednesday, November 9, "Projective dimension and the Frobenius homomorphism", by J. Olmo.
		Wednesday, November 16, "Projective dimension and the Frobenius homomorphism, II", by J. Olmo.
		Wednesday, November 30, "Projective dimension and the Frobenius homomorphism, III", by J. Olmo.
	www.math.sc.edu /~kustin/seminar/Fall2005.html (306 words)

	libecc: Elliptic Curve Cryptography C++ Library - Reference Manual (Site not responding. Last check: 2007-11-01)
		An automorphism is just an isomorphism of a system of objects onto itself.
		The Frobenius homomorphism is a map named after the mathematician Frobenius, and its just 'cause he was born first that it has his name because all he did was realize that for a field with characteristic p we have (a + b)
		Indeed, the Frobenius homomorphism for our field is defined as:
	libecc.sourceforge.net /reference-manual/group__theory__frobenius.html (296 words)

	ABSTRACT ALGEBRA: OnLine Study Guide, §8.1 Solved Problems
		Determine the group of all automorphisms of a field with 4 elements.
		Any automorphism of F must leave 0 and 1 fixed, so the only possibility for an automorphism other than the identity is to interchange a and 1+a.
		Thus the function that fixes 0 and 1 while interchanging a and 1+a is in fact the Frobenius automorphism of F. Next problem
	www.math.niu.edu /~beachy/abstract_algebra/study_guide/soln81.html (527 words)

	FrobeniusAutomorphism (Site not responding. Last check: 2007-11-01)
		automorphism, called the Frobenius automorphism, or simply the Frobenius map on
		Note: This morphism is sometimes also called the ``small Frobenius'' to distinguish it from the map
		This map is then also referred to as the ``big Frobenius'' or the ``power Frobenius map''.
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/entries/FrobeniusAutomorphism/FrobeniusAutomorphism.html (62 words)

	GAP Manual: 18 Finite Fields (Site not responding. Last check: 2007-11-01)
		Note that the subfield over which a field was constructed determines over which field the Galois group, conjugates, norm, trace, minimal polynom, and characteristic polynom are computed (see GaloisGroup, Conjugates, Field Functions for Finite Fields).
		The Galois group of a finite field F of size p^m over a subfield S of size q = p^n is a cyclic group of size m/n.
		It is generated by the Frobenius automorphism that takes every element of F to its q-th power.
	www.mcs.kent.edu /system/documentation/gap/CHAP018.htm (1687 words)

	Element Operations
		There is an automorphism which maps the elementary symmetric function to the homogeneous symmetric function.
		When the power sum symmetric functions are involved, it may be necessary to work with a coefficient ring which allows division by an integer.
		It is known that the Frobenius automorphism on the Schur functions acts just by conjugating the indexing partitions.
	modular.fas.harvard.edu /docs/magma/htmlhelp/text1333.htm (1646 words)

	Titles and Abstracts of MAGC Talks
		3) The role of the Frobenius automorphism in cryptography
		We show how to derive a fast group operation by using the Frobenius automorphism and give evidence that there are many groups obtainable having almost prime order.
		We explain how the Frobenius automorphism is used and give details on the involved algorithms.
	www.iccip.csl.uiuc.edu /conf/magc/2000/titles.html (764 words)

	Cyclic Codes and the Frobenius Automorphism (ResearchIndex)
		Abstract: Using the theory of cyclic codes some results connected with the Frobenius automorphism are proved.
		So the following problem of K.Burde's on characterizing finite fields GF (q n) is solved: Consider GF (q n) as a vector space over GF (q).
		For which GF (q n) exists for any k = 0; : : : ; n exactly one subspace C with dim C = k and which is invariant under the Frobenius automorphism.
	citeseer.ist.psu.edu /winterhof97cyclic.html (317 words)

	[No title]
		The "covering space" philosophy still applies, since we can define what it means for a graph G' to be a covering space of a graph G. Any prime loop P in G defines a deck transformation of G'.
		The reason is that people like to express the zeta function of a discrete dynamical system f: X -> X in terms of the number of fixed points of f^n.
		When f is the Frobenius automorphism, these are usually called "points defined over the field with p^n elements".
	math.ucr.edu /home/baez/twf_ascii/week216 (1888 words)

	ZFR: Frobenius Automorphism (Site not responding. Last check: 2007-11-01)
		This program reads a matrix, applies the Frobenius automorphism, which raises an element x to the p-th power, where p is the characteristic of the field, to each entry and writes out the result.
		Default file names are G1 for input and P2 for output.
		Then the matrix is read row by row, the Frobenius automorphism is applied to each entry, and the resulting row is written out.
	www.math.rwth-aachen.de /homes/MTX/htmldoc/node30.html (114 words)

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