HW2 Solutions
... By Proposition 13 on page 309 of Dummit and Foote it is irreducible in the polynomial ring Z[x]. (Take P to be the prime ideal (3) of Z.) But it follows from Gauss’ Lemma (Proposition 5 on page 303) that if it can be factored in Q[x] then it can be factored in Z[x], which it can’t be, so it is irred ...
... By Proposition 13 on page 309 of Dummit and Foote it is irreducible in the polynomial ring Z[x]. (Take P to be the prime ideal (3) of Z.) But it follows from Gauss’ Lemma (Proposition 5 on page 303) that if it can be factored in Q[x] then it can be factored in Z[x], which it can’t be, so it is irred ...
Algebraic Transformation Groups and Algebraic Varieties
... classifying affine G/H by means of its internal geometric structure as a fiber bundle. Cohomological characterizations of affine G/H provide useful vanishing theorems and related information if one already knows G/H is affine. Such characterizations cannot be realistically applied to prove that a given hom ...
... classifying affine G/H by means of its internal geometric structure as a fiber bundle. Cohomological characterizations of affine G/H provide useful vanishing theorems and related information if one already knows G/H is affine. Such characterizations cannot be realistically applied to prove that a given hom ...
Construction X for quantum error
... in C. As is well known, wt(C) also equals the minimum Hamming distance of any two distinct codewords in C. A quantum error-correcting code (or just quantum code) is a code that protects quantum information from corruption by noise (decoherence) on the quantum channel in a way that is similar to how ...
... in C. As is well known, wt(C) also equals the minimum Hamming distance of any two distinct codewords in C. A quantum error-correcting code (or just quantum code) is a code that protects quantum information from corruption by noise (decoherence) on the quantum channel in a way that is similar to how ...
Fibre Bundles and Homotopy Exact Sequence
... U ⊂ G/H is open if and only if p−1 (G/H) is open in G. G/H with this push–forward topology is a Hausdorff space and even a C ∞ -manifold where p : G → G/H is a C ∞ -map. Check that (G, G/H, H, p) is a fibre bundle. A generalisation of this example is a bundle (E, E/G, G, p) obtained as a factorisati ...
... U ⊂ G/H is open if and only if p−1 (G/H) is open in G. G/H with this push–forward topology is a Hausdorff space and even a C ∞ -manifold where p : G → G/H is a C ∞ -map. Check that (G, G/H, H, p) is a fibre bundle. A generalisation of this example is a bundle (E, E/G, G, p) obtained as a factorisati ...
MAXIMAL REPRESENTATION DIMENSION FOR GROUPS OF
... no group of order 27 has these properties. From Table I we can determine which groups G (up to isoclinism, cf. Remark 6) have |C(G)| = 8 and using Table II we can determine m1 and m2 for these groups. There is no group G with |C(G)| = 8 and m1 + 4m2 = 16. ...
... no group of order 27 has these properties. From Table I we can determine which groups G (up to isoclinism, cf. Remark 6) have |C(G)| = 8 and using Table II we can determine m1 and m2 for these groups. There is no group G with |C(G)| = 8 and m1 + 4m2 = 16. ...
Chapter 3, Groups
... If we look at Propositions 1.9 and 2.14, we can see that the composition of maps in Sn and the sum in Z/nZ have some common properties: Both operations are associative, have an identity element (Id for the composition of maps, 0 for the sum), and every element has an inverse. It turns out that it is ...
... If we look at Propositions 1.9 and 2.14, we can see that the composition of maps in Sn and the sum in Z/nZ have some common properties: Both operations are associative, have an identity element (Id for the composition of maps, 0 for the sum), and every element has an inverse. It turns out that it is ...
Sets Math 130 Linear Algebra
... For instance, we have arithmetic on R, so solutions to equations are subsets of R. The solutions to the equation x3 = x are 0, 1, and −1. We can Examples. Let’s start with sets of numbers. describe its solution set using the notation There are ways of constructing these sets, but let’s S = {x ∈ R | ...
... For instance, we have arithmetic on R, so solutions to equations are subsets of R. The solutions to the equation x3 = x are 0, 1, and −1. We can Examples. Let’s start with sets of numbers. describe its solution set using the notation There are ways of constructing these sets, but let’s S = {x ∈ R | ...
Quotients of adic spaces by finite groups
... Let w be any extension of v to L. Since L/K is a finite algebraic extension, Γw /Γv is a finite group. In particular, w(�) < 1 is cofinal for Γw . Now choose some element x ∈ L with L = K[x]. For large n � 0, the element �n x ∈ L◦ will satisfy a monic polynomial with coefficients in K ◦ , and then L0, ...
... Let w be any extension of v to L. Since L/K is a finite algebraic extension, Γw /Γv is a finite group. In particular, w(�) < 1 is cofinal for Γw . Now choose some element x ∈ L with L = K[x]. For large n � 0, the element �n x ∈ L◦ will satisfy a monic polynomial with coefficients in K ◦ , and then L0, ...
211 - SCUM – Society of Calgary Undergraduate Mathematics
... of w and θ is the angle by which it is rotated in the plane. ...
... of w and θ is the angle by which it is rotated in the plane. ...
8. Group algebras and Hecke algebras
... By determining which εH (hgi ) εH equal to each other each other we may write out the product as unique linear combination of Hecke basis elements. We summarize this discussion as the following. Proposition 8.5 The Hecke algebra is an associative algebra with basis εH hj εH , in 1-1 correspondence t ...
... By determining which εH (hgi ) εH equal to each other each other we may write out the product as unique linear combination of Hecke basis elements. We summarize this discussion as the following. Proposition 8.5 The Hecke algebra is an associative algebra with basis εH hj εH , in 1-1 correspondence t ...
Basis (linear algebra)
Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference.A set of vectors in a vector space V is called a basis, or a set of basis vectors, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set. In more general terms, a basis is a linearly independent spanning set.Given a basis of a vector space V, every element of V can be expressed uniquely as a linear combination of basis vectors, whose coefficients are referred to as vector coordinates or components. A vector space can have several distinct sets of basis vectors; however each such set has the same number of elements, with this number being the dimension of the vector space.