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case and Galois
An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
Waterhouse showed that every profinite group is isomorphic to one arising from the Galois theory of some field K ; but one cannot ( yet ) control which field K will be in this case.
In some cases, it is possible to define abstractions using Galois connections ( α, γ ) where α is from L to L ′ and γ is from L ′ to L. This supposes the existence of best abstractions, which is not necessarily the case.
However, the Langlands correspondence does not include as much arithmetical information about finite Galois extensions as class field theory does in the abelian case.
In that case results have been obtained for p-adic L-functions, which describe certain Galois modules.
In this case the form of the solutions depends on the structure of this Galois group.
Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.
For example, let P be an irreducible polynomial with integer coefficients and p be a prime number which does not divides the leading coefficient of P. Let Q be the polynomial over the finite field with p elements, which is obtained by reducing modulo p the coefficients of P. Then, if Q is separable ( which is the case for every p but a finite number ) then the degrees of the irreducible factors of Q are the lengths of the cycles of some permutation of the Galois group of P.
" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.
On the other hand, an arbitrary algebraic extension may not possess an intermediate extension K that is purely inseparable over F and over which E is separable ( however, such an intermediate extension does exist when is a finite degree normal extension ( in this case, K can be the fixed field of the Galois group of E over F )).
In that case a quantitative measure of ramification is defined for Galois extensions, basically by asking how far the Galois group moves field elements with respect to the metric.
When L / K is separable, a formula similar to the Galois case above can be obtained.
A special case that is easier to state says that if K is an algebraic number field which is a Galois extension of Q of degree n, then the prime numbers that completely split in K have density
In this case, the field extension has degree 4 and is abelian, with the Galois group isomorphic to the Klein four-group.
This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K / Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ ( N )/ m, where m is multiplicative order of p modulo N ; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.
The statement of the Chebotarev density theorem can be generalized to the case of an infinite Galois extension L / K that is unramified outside a finite set S of primes of K ( i. e. if there is a finite set S of primes of K such that any prime of K not in S is unramified in the extension L / K ).
In this case, the Galois group G of L / K is a profinite group equipped with the Krull topology.
This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions in the case of infinite degree.
reciprocity map to the case of Galois groups which are extensions of abelian groups by finite groups.
In this case the generalized ideal class group is the ideal class group of K, and the existence theorem says there exists a unique abelian extension L / K with Galois group isomorphic to the ideal class group of K such that L is unramified at all places of K. This extension is called the Hilbert class field.
In fact for a while the multiplicative case of a 1-cocycle for groups that are not necessarily cyclic was formulated as the solubility of Noether's equations, named for Emmy Noether ; they appear under this name in Emil Artin's treatment of Galois theory, and may have been folklore in the 1920s.

case and extension
If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.
In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).
All stations were above ground and were generally unstaffed, unlike extension stations built ground which are required by law to be staffed in case evacuation is needed.
In a non-criminal case in a United States district court, a litigant ( or a litigant's attorney ) who presents any pleading, written motion or other paper to the court is required, under Rule 11 of the Federal Rules of Civil Procedure, to certify that, to the best of the presenter's knowledge and belief, the legal contentions " are warranted by existing law or by a nonfrivolous argument for the extension, modification, or reversal of existing law or the establishment of new law ".
It is named for Hans Hahn and Stefan Banach who proved this theorem independently in the late 1920s, although a special case was proved earlier ( in 1912 ) by Eduard Helly, and a general extension theorem from which the Hahn – Banach theorem can be derived was proved in 1923 by Marcel Riesz.
The extension ψ is in general not uniquely specified by φ, and the proof gives no explicit method as to how to find ψ: in the case of an infinite dimensional space V, it depends on Zorn's lemma, one formulation of the axiom of choice.
* Level 1: File names are limited to eight characters with a three-character extension, using upper case letters, numbers and underscore only.
In such cases, it could also happen that the Khronos Group's Architecture Review Board gives the extension their explicit approval, in which case the identifier ARB is used.
In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module.
Nevertheless, the Keywell brothers ( and by extension the Purple Gang ) would remain ensnared in the massacre case for all time.
An extension of such a theory to the general case of the L-subsets is possible ( see Gerla 2006 ).
As an extension of the case of multi-valued logic, valuations () of propositional variables () into a set of membership degrees () can be thought of as membership functions mapping predicates into fuzzy sets ( or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation ).
There are two sensible possibilities: either the data are even about the sample a, in which case the even extension is dcbabcd, or the data are even about the point halfway between a and the previous point, in which case the even extension is dcbaabcd ( a is repeated ).
The simplest extension of the unmagnetized case is simply to use the projected area rather than the surface area of the electrode.
Glenn Gould states that variations 28 and 29 present the only case of " motivic collaboration or extension between successive variations.
Among them was an explanation of why the natives of Krypton perished if they had possessed super powers on their native world ( as was the case in the earliest versions of Krypton outlined above, although this only became a problem once Superman — and by extension anyone from Krypton — was portrayed as increasingly powerful, able to withstand nuclear explosions, contrasted with his original power level in which a bursting mortar shell could penetrate his skin ).
In the case of the Alps, the extension could only take place in a west – east direction because the Apulian plate was still converging from the south.
Each race — or hand — is usually 700 miles ( or kilometers ) long, but the first player to complete that distance exactly has the option to declare an extension in which case the race becomes 1, 000 miles.

case and L
* If the balance factor of L is + 1, a single right rotation ( with P as the root ) is needed ( Left-Left case ).
For the case of an object that is small compared with the radial distance to its axis of rotation, such as a tin can swinging from a long string or a planet orbiting in a circle around the Sun, the angular momentum can be expressed as its linear momentum,, crossed by its position from the origin, r. Thus, the angular momentum L of a particle with respect to some point of origin is
The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.
Often, the criteria had moral bases, such as in the case of Pierre de La Primaudaye's L ' Académie française and Guillaume Telin's Bref sommaire des sept vertus & c .. Encyclopaedists encountered several problems with this approach, including how to decide what to omit as unnecessary, how to structure knowledge that resisted structure ( often simply as a consequence of the sheer amount of material that deserved inclusion ), and how to cope with the influx of newly discovered knowledge and the effects that it had on prior structures.
The main difference between the GPL and the LGPL is that the latter allows the work to be linked with ( in the case of a library, ' used by ') a non -( L ) GPLed program, regardless of whether it is free software or proprietary software.
Esperanto's inventor, L. L. Zamenhof, had suggested in an 1894 proposition for a Reformed Esperanto several changes that Ido adopted: eliminating the accented letters and the accusative case, changing the plural to an Italianesque-i, and replacing the table of correlatives with more Latinate words.
Based on this observation alone, Paul L. Maier calls the case for the total authenticity of the Testimonium " hopeless ".
# We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L < sub > g </ sub >( h ) = gh.
If the lower case L is used as the symbol, it is sometimes rendered as a cursive ℓ to help distinguish it from the capital " I ", although this usage has no official approval by any international bureau.
In the multi-dimensional case this corresponds to minimizing the L < sub > 2 </ sub > norm.
" Judge Gerard L. Goettel instructed the jury that the case involved the Unification Church, Sun Myung Moon, and his followers, whom the judge stated were " sometimes referred to as Moonies.
Christine Jorgensen, Beth Elliott, Renée Richards, Sandy Stone, Billy Tipton, Alan L. Hart, April Ashley, Caroline Cossey (" Tula "), Jahna Steele, and Nancy Jean Burkholder were outed as transsexuals by European or American media or, in the case of Billy Tipton, by his coroner.
then there is a point ( 0, f )— the focus, F — such that any point P on the parabola will be equidistant from both the focus and the linear directrix, L. The linear directrix is a line perpendicular to the axis of symmetry of the parabola ( in this case parallel to the x axis ) and passes through the point ( 0 ,- f ).
Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea del Centina, “ she had actually shown that it holds for every exponent p < 197 .” L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.
In Dorothy L. Sayers's novel Gaudy Night, set in 1935, the main character Harriet Vane, a crime fiction writer, covers her investigation on a mystery case at her fictional Oxford college, Shrewsbury, with research on Sheridan Le Fanu.
In the case of One Spy Too Many, a subplot featuring Yvonne Craig as an U. N. C. L. E.
In Rosen v. United States ( 1896 ), the Supreme Court adopted the same obscenity standard as had been articulated in a famous British case, Regina v. Hicklin, L. R. 3 Q.
The riots were a result of the state court acquittal of three White and one Hispanic L. A. police officers by an all-white jury in a police brutality case involving motorist Rodney King, but in 1993, all four officers were convicted in a federal civil rights case.

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