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Page "Splitting field" ¶ 6
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Given and separable
* Given a field K, the multiplicative group ( K < sup > s </ sup >)< sup >×</ sup > of a separable closure of K is a Galois module for the absolute Galois group.
Given a finite separable field extension L / K and a torus T over L, we have a Galois module isomorphism

Given and extension
Given a subset X of a manifold M and a subset Y of a manifold N, a function f: X → Y is said to be smooth if for all p in X there is a neighborhood of p and a smooth function g: U → N such that the restrictions agree ( note that g is an extension of f ).
Given the existence as uttered forth in the public works of Puncher and Wattmann of a personal God quaquaquaqua with white beard quaquaquaqua outside time without extension who from the heights of divine apathia divine athambia divine aphasia loves us dearly with some exceptions for reasons unknown but time will tell and suffers like the divine Miranda with those who for reasons unknown but time will tell are plunged in torment ...
; Degree of an extension: Given an extension E / F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by: F.
; Generating set: Given a field extension E / F and a subset S of E, we write F ( S ) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +,-,*,/ on the elements of F and S. If E = F ( S ) we say that E is generated by S over F.
; Inverse problem of Galois theory: Given a group G, find an extension of the rational number or other field with G as Galois group.
Given a language L, and a pair of strings x and y, define a distinguishing extension to be a string z such that
Given the empty string, 00 ( or 11 ), 01 and 10 are distinguished extensions resulting in the three classes ( corresponding to numbers that give remainders 0, 1 and 2 when divided by 3 ), but after this step there is no distinguished extension anymore.
Given then a normal extension L of K, with automorphism group Aut ( L / K ) = G, and containing α, any element g ( α ) for g in G will be a conjugate of α, since the automorphism g sends roots of p to roots of p. Conversely any conjugate β of α is of this form: in other words, G acts transitively on the conjugates.
Given a field extension K / F, the field K can be considered as a vector space over the field F. The dimension of this vector space is the degree of the extension and is denoted by: F.
Given additionally a map, one says that has the homotopy lifting extension property if:
Given an exterior connection D satisfying this compatibility property, there exists a unique extension of D:
Given R-modules A and B, an extension of A by B is a short exact sequence of R-modules
Given their newly strengthened bargaining position, the mercenaries vastly inflated their original demands, even requiring the extension of the payments to the Libyans whom Carthage had conscripted ( and who were not mercenaries ) as well as other Numidians and to the escaped slaves and the like who had joined their ranks against Carthage.
Given an unramified finite extension L / K of local fields, there is a concept of Frobenius endomorphism which induces the Frobenius endomorphism in the corresponding extension of residue fields.
* Given an extension as above, it is unramified in all but finitely many points.
Given any real vector space V we may define its complexification by extension of scalars:
Given some mereotopology X, adding C8 to X results in what Casati and Varzi call the Whiteheadian extension of X, denoted WX.
Given any central extension of groups

Given and K
Given two groups G and H and a group homomorphism f: G → H, let K be a normal subgroup in G and φ the natural surjective homomorphism G → G / K ( where G / K is a quotient group ).
Given a field K, the corresponding general linear groupoid GL < sub >*</ sub >( K ) consists of all invertible matrices whose entries range over K. Matrix multiplication interprets composition.
Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X.
Given any vector space V over K we can construct the tensor algebra T ( V ) of V. The tensor algebra is characterized by the fact:
A Clifford algebra Cℓ ( V, Q ) is a unital associative algebra over K together with a linear map satisfying for all defined by the following universal property: Given any associative algebra A over K and any linear map such that
Given a vector space V over a field K, the span of a set S ( not necessarily finite ) is defined to be the intersection W of all subspaces of V which contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W.
* Given any morphism k : K → X such that f k is the zero morphism, there is a unique morphism u: K K such that k u
Given a locally compact topological field K, an absolute value can be defined as follows.
Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.
Given a set S with three subsets, J, K, and L, the following holds:
Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generated by the roots of p. If K is a subfield of the complex numbers, the existence is automatic.

Given and
Given the cost benefits and personal enjoyment of boat building, do-it-yourself Kit Boats were also introduced using plywood material.
Given a DMS ( Degrees, Minutes, Seconds ) coordinate such as W87 ° 43 41 ″, convert it to a number of decimal degrees using the following method:
The Alaska Time Zone applies to the territory of the state of Alaska east of 169 ° 30 W. Given that the UTC − 9 time corresponds to the solar time at 9 × 15 ° = 135 ° W ( roughly, Juneau ), the westernmost locales of the Alaska Time Zone are off by up to 169 ° 30 − 135 ° = 34 ° 30 from local solar time.
Given a standard training set D of size n, bagging generates m new training sets, each of size n < n, by sampling examples from D uniformly and with replacement.
Given a fiber bundle π: E → B and a continuous map f: B → B one can define a " pullback " of E by f as a bundle f * E over B ′.
Given an arbitrary Riemannian metric g on an almost complex manifold M one can construct a new metric g compatible with the almost complex structure J in an obvious manner:

Given and Galois
Given, a representation of on a finite-dimensional complex vector space, where is the Galois group of the finite extension of number fields, the Artin-function: is defined by an Euler product.

Given and closure
So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities ( the axioms ), one may consider their symmetric, transitive closure E. The quotient algebra T / E is then the algebraic structure or variety.
Given their sting, however, they must always be treated with caution, and the discovery of men o ' war washed up on a beach may lead to the closure of the whole beach.
Given a graph G with n vertices, the closure cl ( G ) is uniquely constructed from G by repeatedly adding a new edge uv connecting a nonadjacent pair of vertices u and v with until no more pairs with this property can be found.
Given this closure property for CSAs, they form a monoid under tensor product, compatible with Brauer equivalence, and the Brauer classes are all invertible: the inverse class to that of an algebra A is the one containing the opposite algebra A < sup > op </ sup > ( the opposite ring with the same action by K since the image of K → A is in the center of A ).
# Given that the system is in a correct state, it is guaranteed to stay in a correct state, provided that no fault happens ( closure ).
Given an operator T, the range of the continuous functional calculus h → h ( T ) is the ( abelian ) C *- algebra C ( T ) generated by T. The Borel functional calculus has a larger range, that is the closure of C ( T ) in the weak operator topology, a ( still abelian ) von Neumann algebra.
Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure.
Given the celebrations of 40 years of Maltese independence, the 25 years of the closure of the British Base in Malta, and the anticipated admission of Malta into the European Union, the three Lodges of the Irish Constitution met on 5 September 2003 and resolved to form themselves into a Sovereign Grand Lodge.
Given a linear operator, not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of, and we say that is closable.

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