Henry Cohn`s home page
... is). If α is the root of an irreducible polynomial of degree n, then K(α)/K is finite, and has degree n. Proof. Consider the map π : K[x] → K[α] that substitutes α for x in a polynomial. The kernel of this map is a principal ideal (since K[x] is a PID), generated by a polynomial f , which is the pol ...
... is). If α is the root of an irreducible polynomial of degree n, then K(α)/K is finite, and has degree n. Proof. Consider the map π : K[x] → K[α] that substitutes α for x in a polynomial. The kernel of this map is a principal ideal (since K[x] is a PID), generated by a polynomial f , which is the pol ...
On Graphs with Exactly Three Q-main Eigenvalues - PMF-a
... s Q-eigenvalues qk for k = 1, 2, . . . , s if and only if the vectors j, Qj, . . . , Qs−1 j are linearly independant and [ sk=1 (Q − qk I)] j = 0. In [6, 8] all the graphs with exactly two Q-main eigenvalues are characterized. Theorem 2.3 ([6]). A graph G has exactly two Q-main eigenvalues if and on ...
... s Q-eigenvalues qk for k = 1, 2, . . . , s if and only if the vectors j, Qj, . . . , Qs−1 j are linearly independant and [ sk=1 (Q − qk I)] j = 0. In [6, 8] all the graphs with exactly two Q-main eigenvalues are characterized. Theorem 2.3 ([6]). A graph G has exactly two Q-main eigenvalues if and on ...
8 The Gelfond-Schneider Theorem and Some Related Results
... Observe that αs1 +s2 β is an algebraic number for all integers s1 and s2 . To establish the theorem, it suffices to show that there are two distinct pairs of integers (s1 , s2 ) and (s01 , s02 ) for which s1 + s2 β = s01 + s02 β. We will choose S sufficiently large and show such pairs exist with 0 ≤ ...
... Observe that αs1 +s2 β is an algebraic number for all integers s1 and s2 . To establish the theorem, it suffices to show that there are two distinct pairs of integers (s1 , s2 ) and (s01 , s02 ) for which s1 + s2 β = s01 + s02 β. We will choose S sufficiently large and show such pairs exist with 0 ≤ ...
Representation schemes and rigid maximal Cohen
... R-modules. Furthermore for any point M : A → EndR (V• ⊗ R) in RepR (A, V• ), there is an exact sequence of k-vector spaces 0 → EndA (M )0 → EndR (V• ⊗ R)0 → TM RepR (A, V• ) → Ext1A (M, M )0 → 0, where and TM RepR (A, V• ) is the Zariski tanget space to the scheme RepR (A, V• ) at the point M (see T ...
... R-modules. Furthermore for any point M : A → EndR (V• ⊗ R) in RepR (A, V• ), there is an exact sequence of k-vector spaces 0 → EndA (M )0 → EndR (V• ⊗ R)0 → TM RepR (A, V• ) → Ext1A (M, M )0 → 0, where and TM RepR (A, V• ) is the Zariski tanget space to the scheme RepR (A, V• ) at the point M (see T ...
Principal bundles on the projective line
... homogeneous space over Spec k. In this article, we show Main theorem. Let E → P1k be a principal G-bundle on P1k which is trivial at the origin. Then E is isomorphic to Eλ,G for some one-parameter subgroup λ : Gm → G defined over k. In particular, since every one-parameter subgroup lands inside a ma ...
... homogeneous space over Spec k. In this article, we show Main theorem. Let E → P1k be a principal G-bundle on P1k which is trivial at the origin. Then E is isomorphic to Eλ,G for some one-parameter subgroup λ : Gm → G defined over k. In particular, since every one-parameter subgroup lands inside a ma ...
m-Ary Hypervector Space: Convergent Sequence and Bundle Subsets
... both with an internal hyperoperation and an external hyperoperation is the so-called hypermodule. n-Ary generalizations of algebraic structures is the most natural way for further development and deeper understanding of their fundamental properties. The notion of n-ary group was introduced by Dörnt ...
... both with an internal hyperoperation and an external hyperoperation is the so-called hypermodule. n-Ary generalizations of algebraic structures is the most natural way for further development and deeper understanding of their fundamental properties. The notion of n-ary group was introduced by Dörnt ...
CONVERGENCE THEOREMS FOR PSEUDO
... x0 ) → 0, where pB is the norm on EB which has B as its unit ball. It may be remarked that Köthe names this concept ‘local convergence’. Mackey-convergence implies convergence in the topology τ (Webb [10]). Conversely, in a metrizable locally convex space every convergent sequence is Mackey-converg ...
... x0 ) → 0, where pB is the norm on EB which has B as its unit ball. It may be remarked that Köthe names this concept ‘local convergence’. Mackey-convergence implies convergence in the topology τ (Webb [10]). Conversely, in a metrizable locally convex space every convergent sequence is Mackey-converg ...
Locally ringed spaces and affine schemes
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
Notes on Measure Theory Definitions and Facts from Topic 1500
... space, the measure of the union of those disjoint sets is equal to the sum of the measures of the individual sets. • Let A ⊆ 2M . A function µ : A → [0, is finitely F∞] is σ-additive F if µ P additive & ∀pw − dj A1 , ..., An ∈ A, Aj ∈ A ⇒ µ( Aj ) = µ(Aj ). σ-additive differs from finitely additive i ...
... space, the measure of the union of those disjoint sets is equal to the sum of the measures of the individual sets. • Let A ⊆ 2M . A function µ : A → [0, is finitely F∞] is σ-additive F if µ P additive & ∀pw − dj A1 , ..., An ∈ A, Aj ∈ A ⇒ µ( Aj ) = µ(Aj ). σ-additive differs from finitely additive i ...
BANACH ALGEBRAS 1. Banach Algebras The aim of this notes is to
... ρA (a) of a with respect to A is defined by ρA (a) := {λ ∈ C : a − λ1 ∈ G(A)}. The spectrum σA (a) of a with respect to A is defined by σA (a) = C \ ρA (a). That is same as saying σA (a) := {λ ∈ C : a − λ1 is not invertible in A}. If B is a closed subalgebra of A such that 1 ∈ B. If a ∈ B, then once ...
... ρA (a) of a with respect to A is defined by ρA (a) := {λ ∈ C : a − λ1 ∈ G(A)}. The spectrum σA (a) of a with respect to A is defined by σA (a) = C \ ρA (a). That is same as saying σA (a) := {λ ∈ C : a − λ1 is not invertible in A}. If B is a closed subalgebra of A such that 1 ∈ B. If a ∈ B, then once ...
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.