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Topic: Splitting lemma

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	PlanetMath: proof of fundamental theorem of Galois theory
		The theorem is a consequence of the following lemmas, roughly corresponding to the various assertions in the theorem.
		, it is the splitting field of a
		This is version 2 of proof of fundamental theorem of Galois theory, born on 2004-06-23, modified 2004-06-29.
	planetmath.org /encyclopedia/ProofOfFundamentalTheoremOfGaloisTheory.html (296 words)

	Snake lemma -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-22)
		The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity.
		The important statement of the lemma is that a connecting homomorphism d exists which completes the exact sequence.
		In the case of abelian groups or (A self-contained component (unit or item) that is used in combination with other components) modules over some (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring, the map d can be constructed as follows.
	www.absoluteastronomy.com /encyclopedia/S/Sn/Snake_lemma.htm (571 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		As noted in the introductory paragraph such splitting fields always exist, but before stating that formally, we want to establish the essential uniqueness of a splitting field.
		Lemma.} {Let $\phi:k\longrightarrow k'$ be an isomorphism of fields and let $f \in k[X]$.
		Prove that $F$ is a splitting field of $f$ over $k$ iff $F$ is a splitting field of $f$ over $E$.
	darkwing.uoregon.edu /~anderson/math649/lecture23.html (1719 words)

	[No title]
		Also, some more recent results can be seen as natural in the light of the splitting lemma; among these we mention the beautiful work by Krupa [11] on bifurcation from a relative equilibrium, and the very recent contribution by Chossat and Koenig [12].
		Then, $\psi$ induces an autonomous vector field $A$ on $\Om$, and a $\Om$-dependent flow on $G$.} {\bf Splitting Lemma.} (Coordinate formulation) {\it Let $M \sse R^n$ be a smooth manifold, $G$ a smooth compact connected Lie group acting smoothly on $M$.
		In this bifurcation, the symmetry $G_0 \subset G$ corresponding to $\phi$ rotations is broken; the splitting lemma guarantees that the bifurcating flow is given by a small deformation of the flow on $S^1$ times a flow on $G_0$; this is indeed the kind of results obtained by Krupa [11].
	www.ma.utexas.edu /mp_arc/papers/94-318 (2183 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.
		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.
		The next lemma shows that in computing Galois groups it is enough to consider polynomials with integer coefficients.
	www.math.niu.edu /~beachy/aaol/galois.html (1898 words)

	[No title]
		Lemma 1 The ønite diöerence equations (7) and (8) are equivalent.
		Lemma 2 Suppose that u is the grid function deøned on µ!x with zero boundary conditions and that kDkhuk <= C < 1 for all positive h1; h2 <= h0.
		Lemma 6 Assume that v 2 W and (2) is satisøed.
	www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file182.html (5380 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		For a stabilization of a Heegaard splitting, we have: \begin{proposition}\label{prop:stabilization} %Let $\alpha$ be an arc properly embedded in $I \times I$, %such that $\Phi^{-1}(\alpha (I))$ is a Heegaard surface of $M$.
		Let $A \cup_P B$ be a genus 0 Heegaard splitting of $S^3$ which gives a 2-bridge position of $K$, and $X \cup_Q Y$ a genus one Heegaard splitting which gives a genus one 1-bridge position of $K$.
		The argument in the proof of Lemma C-2 works in this case to show that there is an equivariant compressing disk $G$ for $\tilde{F}$ such that $G$ is non-separating, and $G$ intersects $\tilde{\alpha}$ in one point, and this gives the conclusion.
	home.imf.au.dk /esn/preprints/093 (14975 words)

	[No title]
		Splitting fields could be avoided in this example because of the fact that $p_1,p_2,p_3$ were apparent singularities.
		This is a subfield of the splitting field of $a_na_0$.
		Splitting fields over $C_0$ are not needed for this computation.
	www.math.fsu.edu /~hoeij/papers/FiniteSingularities/oud/final_version (9877 words)

	Sets and Models of
		Lemma 4.3 With notation as in Lemma 4.2, for all
		As in the proof of Lemma 3.19 we have that, for all
		This is a uniform relativization of part 2 of Lemma 3.16.
	www.math.psu.edu /simpson/papers/pizowkl (1962 words)

	[No title]
		In [16], a variety of heuristics for splitting the stacks representing a tree are presented.
		The splitting function is able to split a subproblem of size T into two subproblems of size T1 and T2 in unit time.
		A subproblem generated by h subsequent splits of the root problem is guaranteed to be reduced to a constant atomic size Tatomic or smaller.
	www.ubka.uni-karlsruhe.de /vvv/ira/1995/6/6.text (4402 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		An older term for a splitting field is a root field.
		In particular, $K (a_1, a_2, a_3)$ is the splitting field in $\C$ of the cubic polynomial $f(x) \in K [x]$.
		We note that the roots of $f(x)$ are obtained by rational operations and by the extraction of cube and square roots.
	www.mcs.drexel.edu /~rboyer/courses/math534/week5 (3322 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		In this paper, we study the global topology of such maps for $p = 3$ and give various new results, among which are a splitting theorem for manifolds admitting special generic maps into ${\bf R}^3$ and a classification theorem of 4- and 5-dimensional manifolds with free fundamental groups admitting special generic maps into ${\bf R}^3$.
		We show that then the manifold splits into a connected sum of two nonsimply connected manifolds with corresponding fundamental groups both of which admit a special generic map into ${\bf R}^3$.
		Furthermore, when the manifold $M$ is stably parallelizable, the same splitting theorem is valid for all $n$ with $\Theta_{n-1}(\partial \pi) = 0$, where $\Theta_m(\partial \pi)$ is the $h$-cobordism group of oriented homotopy $m$-spheres which bound compact parallelizable $(m+1)$-dimensional manifolds.
	home.imf.au.dk /esn/preprints/090 (7797 words)

	[No title]
		Lemma 4.1 Suppose the connected H-space X is a k(n)-tower of finite type with vn-free homotopy.
		Lemma 8.11, sl* ightly modified (or with n replaced by n + 1), shows that (ii) is the lowest of the re *maining terms.
		Lemma 9.3 For k g(n, m) [replaced by k g(n, m) - 1 if p = 2], Qk\ Jm is * the left P (n)-submodule of Qkspanned by all the Q-allowable monomials (8.1) that* * lie in it and contain an explicit factor wq for some q > m.
	hopf.math.purdue.edu /Boardman-Wilson/BWonPn.txt (15997 words)

	Gloups examples (Site not responding. Last check: 2007-10-22)
		Lemma TH1 corresponds to the first element of the sequence - one is required to prove that 1, the first element, is positive.
		Lemma TH3 corresponds to the third element of the sequence, the first one that is obtained in a systematic manner (the preceding two elements were given by initializations of the corresponding Lustre flows, while this one is the sum of the values of two variables).
		Thus, the 22 lemmas correspond to a proof tree, not to an induction of length 22.
	www-verimag.imag.fr /~mikac/Gloups/Gloups-examples.html (1396 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		\item In the splitting field $E$ of a separable polynomial the elements left fixed by every automorphism of the Galois group of $E$ over $F$ are exactly the elements of $F$.
		Eventually, since the degree of $N$ over $F$ is finite, one of the successive splitting fields must be the entire field $N$.
		\textsc{Artin's Lemma:} \ If $H$ is any finite group of automorphisms of a field $N$, while $K$ is the subfield of all elements invariant under $H$, then the degree $[N:K]$ of $N$ over $K$ does not exceed the order of $H$.
	www.mcs.drexel.edu /~rboyer/courses/math534/week6 (2522 words)

	[No title]
		The chromatic splitting conjecture is actually stronger than that, for it also explains how this splitting occurs, but most of the corollari* es I draw from the chromatic splitting* conjecture only need the splitting itself.
		The chromatic splitting conjecture In this section, we describe Hopkins' chromatic splitting conjecture and de- duce some corollaries of it.
		This is the chromatic splitting conjecture for n = 2.
	hopf.math.purdue.edu /Hovey/chromatic-splitting.txt (8224 words)

	Turning the Oxford-Hachette SGML tape into a human-readable dictionary for DEFI (Site not responding. Last check: 2007-10-22)
		As such, lemmas are not to be confused with headwords: the headword is the entry headword, which gives access to the entry in printed dictionaries.
		In such cases the headword (here law) is taken as lemma, and the noun given as lemma by the dictionary is kept in the HEAD field.
		The various lemma variants are stored into a lem[] array using the AWK split function, and a for-loop then reprints the record as many times as there are variants, with only one lemma variant in each case.
	engdep1.philo.ulg.ac.be /michiels/oh2defi.htm (6227 words)

	E.W. Dijkstra Archive: To hell with "meaningful identifiers"! (EWD 1044)
		Lemma 1 A bag containing 1 and at least one further element is not maximal.
		Lemma 2 For x > 4, a bag containing x is not maximal.
		With the replacement of Lemma 2 and a separate treatment of 4's in the bag by (1), we have eliminated a case analysis at the price of a proof by mathematical induction.
	www.cs.utexas.edu /~EWD/transcriptions/EWD10xx/EWD1044.html (841 words)

	Fall 2006 Lectures on the proof of the Bloch-Kato Conjecture
		The goal of the November lectures (which was achieved) will be to prove that μ is nonzero when X is a splitting variety.
		Briefly, Lemma 5.15 is used to prove 5.18, 6.8, 6.9, 6.13, 6.15 and hence 6.11.
		Theorem (Lemma 5.15 in [MC/l]) M is a summand of M(X) satisfying the above axioms.
	www.math.rutgers.edu /~weibel/motivic2006.html (1452 words)

	Citations: A balanced search tree with O - Levcopoulos, Overmars (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		....O(1) worst case update time by using a global splitting lemma that is based on a pebble game combined with the bucketing technique of Overmars (14] Instead of storing single keys in the leaves of the search tree, each leaf can store a list of several keys.
		and Dietz and Raman [3] were based on a global splitting lemma which guaranteed that each of the recursive substructures would have polylogarithmic size.
		The idea is simple: Split buckets with regular intervals, and spread the resulting work on the top tree incrementally over the updates between the splits.
	citeseer.ist.psu.edu /context/161707/0 (2037 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.
		Zorn's Lemma) to construct an algebraic or separable closure of a given field F, and to show that any two algebraic closures are isomorphic over F. The group of isomorphisms of a separable closure, or more generally of any infinite-dimensional separable extension, can be studied by a generalization of Galois theory to infinite Galois groups.
		``Lemma 3'' in Jacobson 4.7 generalizes to normal extensions with a cyclic Galois group of arbitrary order n (not necessarily prime) over a field F containing the n-th roots of unity (that is, F such that the polynomial X
	abel.math.harvard.edu /~elkies/M250.04 (12646 words)

	Roberto Maieli (Site not responding. Last check: 2007-10-22)
		It is shown here that Focusing can also be interpreted in the proof-net formalism, where it appears, at least in the multiplicative fragment, to be a simple refinement of the "Splitting lemma" for proof-nets.
		This change of perspective allows to generalize the Focusing result to (the multiplicative fragment of) any logic where the "Splitting lemma" holds.
		This is, in particular, the case of the Abrusci and Ruet's Non-Commutative logic, and all the computational exploitation of Focusing which has been performed in the commutative case can thus be revised and adapted to the non commutative case.
	logica.uniroma3.it /~maieli/ab-lpar99.html (183 words)

	3. Row-minimal matrices
		(3.1) In virtue of Lemma 2 we shall look for presentation matrices which have a good chance of being equivalent to a matrix in block diagonal form.
		It is possible to check by a standard basis computation whether a row-minimal matrix Q is equivalent to a block diagonal matrix.
		Again, by Lemma 1(i3), we have a factorization
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/15/paper_html/node3.html (244 words)

	Finite Fields
		The proof of the lemma follows directly from the division algorithm.
		Given the previous two lemmas we get the following theorem.
		Since every polynomial has a splitting field, there must be some extension
	www-math.cudenver.edu /~rrosterm/crypt_proj/node4.html (156 words)

	Elementary catastrophe theory: an introduction
		In the fifties it was shown by among others Thom, that the complete state function can be split up into two components, one major, "Morse" component related to the stable areas, one minor, degenerate component, whose number of variables equals the corank of the function.
		The "splitting lemma" has a very central position within the theory: if we have a thousand variables in a function of corank 2 we only need to study a function with 2 variables in order to learn about the behaviour of the function in the vicinity of the degenerate critical point.
		Both the Morse and the Splitting lemma can be extended to families of functions.
	home.swipnet.se /~w-48087/faglar/materialmapp/teorimapp/ekt1.html (4230 words)

	Local Steiner Improvement
		The following lemma provides a way to select specific neighbors which might be found on the path to an optimum tree.
		With this lemma we now have the mechanism necessary to transform one tree into any other by selecting a sequence of transformations leading from that 1-1 tree to the one we wish to construct.
		This means that by restricting our field of inquiry to 1-1 Steiner trees it is possible to construct an optimum solution to the rectilinear Steiner spanning problem by traversing a sequence of transformations between neighboring 1-1 trees, starting from any tree.
	www.cs.engr.uky.edu /~lewis/research/Papers/Steiner-Local/local.html (4174 words)

	Citations: Making data structures persistent - Driscoll, Sarnak, Sleator, Tarjan (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		Levcopoulos and Overmars (13] presented an algorithm achieving O(1) worst case update time by using a global splitting lemma that is based....
		Levcopoulos and Overmars (13] presented an algorithm achieving O(1) worst case update time by using a global splitting lemma that is based on a pebble game combined with the....
		These are precisely the distinct colors of the points contained in [a; b] c; d] e; 1) The query time follows from Lemma 1.3.
	citeseer.ist.psu.edu /context/109463/0 (1585 words)

	CiteULike: New aspects of the ddc-lemma (Site not responding. Last check: 2007-10-22)
		Similarly to the standard dd^c-lemma, its generalized version induces a decomposition of the cohomology of a manifold and causes the degeneracy of the spectral sequence associated to the splitting d = \del + \delbar at E_1.
		But, in contrast with the dd^c-lemma, its generalized version is not preserved by symplectic blow-up or blow-down (in the case of a generalized complex structure induced by a symplectic structure) and does not imply formality.
		We study implications of the `dd^c-lemma' in the generalized complex setting.
	www.citeulike.org /user/allan/article/997612 (299 words)

	Math 250: Higher Algebra (2001-2)
		Indeed f is separable (since f'=-1), and its q roots in a splitting field are closed under the field operations (cf.
		You should already be familiar with, or willing to quickly learn on your own, the material in 1.1 (basic definitions, Schur's lemma, etc.), 1.2 (characters and their orthogonality), and 1.3.4 (the group algebra, introduced in 250a).
		In the proof of Lemma 3.35 (page 40), it might not be obvious that lambda is a positive real.
	abel.math.harvard.edu /~elkies/M250.01 (9028 words)

	Singular Manual: morsesplit
		Normal form of f in M^3 after application of the splitting lemma
		apply the splitting lemma (generalized Morse lemma) to f
		User manual for Singular version 3-0-0, May 2005, generated by
	www.math.lsu.edu /singular/sing_822.htm (64 words)

	Approximating the Logarithm of a Matrix to Specified Accuracy
		A transformation-free form of this method that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented.
		The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products.
		The number of square root stages and the degree of the final Padé approximation are chosen to minimize the computational work.
	epubs.siam.org /sam-bin/dbq/article/36401 (217 words)

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