Notes on Galois Theory
... F [x]. Suppose that α1 , . . . , αn are the (distinct) roots of f (x) that lie in E, i.e. {α ∈ E : f (α) = 0} = {α1 , . . . , αn } and, for i 6= j, αi 6= αj . Then Gal(E/F ) acts on the set {α1 , . . . , αn }, and hence there is a homomorphism ρ : Gal(E/F ) → Sn , where Sn is the symmetric group on ...
... F [x]. Suppose that α1 , . . . , αn are the (distinct) roots of f (x) that lie in E, i.e. {α ∈ E : f (α) = 0} = {α1 , . . . , αn } and, for i 6= j, αi 6= αj . Then Gal(E/F ) acts on the set {α1 , . . . , αn }, and hence there is a homomorphism ρ : Gal(E/F ) → Sn , where Sn is the symmetric group on ...
Computing the p-Selmer Group of an Elliptic Curve
... elliptic curve, but we will need to perform arithmetic in the elds of de nition of the points of order p. Our method shall be based on that found in [15], to which we refer the reader for further details and proofs of some of the results. That paper gives a general algorithm for computing Selmer g ...
... elliptic curve, but we will need to perform arithmetic in the elds of de nition of the points of order p. Our method shall be based on that found in [15], to which we refer the reader for further details and proofs of some of the results. That paper gives a general algorithm for computing Selmer g ...
STRATIFICATION BY THE LOCAL HILBERT
... there exists a unique (q1 , . . . , qr , R) ∈ Rr+1 such that: (i) f = q1 g1 + · · · + qr gr + R (ii) for any j, if qj 6= 0 then Supp(qj ) + exp(gj ) ⊂ ∆j ...
... there exists a unique (q1 , . . . , qr , R) ∈ Rr+1 such that: (i) f = q1 g1 + · · · + qr gr + R (ii) for any j, if qj 6= 0 then Supp(qj ) + exp(gj ) ⊂ ∆j ...
Inversion of Circulant Matrices over Zm
... The above results show that the inversion of the global maps F and G is equivalent to the inversion of A(x) in Zm{x} and Zm [x]/(xn −1) respectively. Therefore they are also equivalent to the inversion of bi-infinite Toeplitz and circulant matrices. Conditions for invertibility over Zm{x} and Zm [x] ...
... The above results show that the inversion of the global maps F and G is equivalent to the inversion of A(x) in Zm{x} and Zm [x]/(xn −1) respectively. Therefore they are also equivalent to the inversion of bi-infinite Toeplitz and circulant matrices. Conditions for invertibility over Zm{x} and Zm [x] ...
Algebras and Representations
... such that (xy)z = x(yz). The algebra A is said to have a unit element if there exists e ∈ A such that ae = ea = a for all a ∈ A. If A has an unit element it is unique and it will usually be denoted by 1. Examples 1. Let V be a vector space over C (possibly infinite dimensional), and let A = End(V ) ...
... such that (xy)z = x(yz). The algebra A is said to have a unit element if there exists e ∈ A such that ae = ea = a for all a ∈ A. If A has an unit element it is unique and it will usually be denoted by 1. Examples 1. Let V be a vector space over C (possibly infinite dimensional), and let A = End(V ) ...
Algebra 1
... You can model the probabilities found in the Punnett square with the expression ( 1 B + 1 W)2. Show that this product gives the same result ...
... You can model the probabilities found in the Punnett square with the expression ( 1 B + 1 W)2. Show that this product gives the same result ...
