ModernCrypto2015-Session12-v2
... A set S endowed with one or more finitary operations is called an algebraic structure. Let S be a set, and : SSS be a binary operation. The pair (S, ) is called a group-like structure. Depending on the properties that satisfies on S, the structure is called by various names (semicategory, ...
... A set S endowed with one or more finitary operations is called an algebraic structure. Let S be a set, and : SSS be a binary operation. The pair (S, ) is called a group-like structure. Depending on the properties that satisfies on S, the structure is called by various names (semicategory, ...
RINGS OF INTEGER-VALUED CONTINUOUS FUNCTIONS
... Our notation is largely taken from (or patterned after) [2]. As usual, Z represents the ring of integers and Q denotes the field of rationals. For any nEZ, let Z„ be the residue class ring of Z modulo the ideal generated by n. Following [2], if nEZ, the symbol n represents the constant function (on ...
... Our notation is largely taken from (or patterned after) [2]. As usual, Z represents the ring of integers and Q denotes the field of rationals. For any nEZ, let Z„ be the residue class ring of Z modulo the ideal generated by n. Following [2], if nEZ, the symbol n represents the constant function (on ...
Ring Theory
... integers modulo 2, and i is such that i2 = −1 ≡ 1 mod 2. This is thus the ring of 2 × 2 matrices with coefficients in F2 [i] = {a + ib, a, b ∈ {0, 1}}. Let I be the subset of matrices with coefficients taking values 0 and 1 + i only. It is a two-sided ideal of M2 (F2 [i]). Indeed, take a matrix U ∈ ...
... integers modulo 2, and i is such that i2 = −1 ≡ 1 mod 2. This is thus the ring of 2 × 2 matrices with coefficients in F2 [i] = {a + ib, a, b ∈ {0, 1}}. Let I be the subset of matrices with coefficients taking values 0 and 1 + i only. It is a two-sided ideal of M2 (F2 [i]). Indeed, take a matrix U ∈ ...
Factoring in Skew-Polynomial Rings over Finite Fields
... rings most generally allow both an automorphism σ of F and a derivation δ : F → F, a linear function such that δ(ab) = σ(a)δ(b) + δ(a)b for any a, b ∈ F. The skew-polynomial ring F[x; σ, δ] is then defined such that xa = σ(a)x + δ(a) for any a ∈ F. In this paper we only consider the case when δ = 0 ...
... rings most generally allow both an automorphism σ of F and a derivation δ : F → F, a linear function such that δ(ab) = σ(a)δ(b) + δ(a)b for any a, b ∈ F. The skew-polynomial ring F[x; σ, δ] is then defined such that xa = σ(a)x + δ(a) for any a ∈ F. In this paper we only consider the case when δ = 0 ...
Commutative Algebra Notes Introduction to Commutative Algebra
... This follows since the gcd of any two numbers can always be represented as an integer combination of the two numbers; a ∩ b is the ideal generated by their lcm (proof easy); and ab = (mn) (proof easy) Thus it follows that ab = a ∩ b ⇐⇒ m, n are coprime. 2. A = k[x1 , . . . , xn ], a = (x1 , . . . , ...
... This follows since the gcd of any two numbers can always be represented as an integer combination of the two numbers; a ∩ b is the ideal generated by their lcm (proof easy); and ab = (mn) (proof easy) Thus it follows that ab = a ∩ b ⇐⇒ m, n are coprime. 2. A = k[x1 , . . . , xn ], a = (x1 , . . . , ...
Finite group schemes
... In other words, every morphism X → Y of k-schemes, with Y /k étale, factors uniquely via X → $0 (X). To understand what $0 (X) is, we use our description of étale k-schemes. Fix a separable closure k̄/k. Observe that Gal(k̄/k) acts on Spec(k̄), hence on, Xk̄ = X ×k Spec(k̄), hence on the topologic ...
... In other words, every morphism X → Y of k-schemes, with Y /k étale, factors uniquely via X → $0 (X). To understand what $0 (X) is, we use our description of étale k-schemes. Fix a separable closure k̄/k. Observe that Gal(k̄/k) acts on Spec(k̄), hence on, Xk̄ = X ×k Spec(k̄), hence on the topologic ...
Separable extensions and tensor products
... 5. Separability for infinite extensions When L/K is an algebraic extension of possibly infinite degree, here is the way separability is defined. Definition 5.1. An algebraic extension L/K is called separable if every finite subextension of L/K is separable. Equivalently, L/K is separable when every ...
... 5. Separability for infinite extensions When L/K is an algebraic extension of possibly infinite degree, here is the way separability is defined. Definition 5.1. An algebraic extension L/K is called separable if every finite subextension of L/K is separable. Equivalently, L/K is separable when every ...
THE EXPONENT THREE CLASS GROUP PROBLEM FOR SOME
... Theorem 10 (see also [Lou4, Theorem 13]). 1. Assume that the exponent of the ideal class group of a simplest cubic field Km is equal to 2. Then fm is a prime equal to 1 mod 6 and p(fm −1)/3 6≡ 1 (mod fm ) for all the primes p such that p2 < 2m + 3. 2. There are only 5 simplest cubic fields Km with i ...
... Theorem 10 (see also [Lou4, Theorem 13]). 1. Assume that the exponent of the ideal class group of a simplest cubic field Km is equal to 2. Then fm is a prime equal to 1 mod 6 and p(fm −1)/3 6≡ 1 (mod fm ) for all the primes p such that p2 < 2m + 3. 2. There are only 5 simplest cubic fields Km with i ...
Theorem 1. Every subset of a countable set is countable.
... We draw attention to a simple principle, which can be used to prove many of the usual and important theorems on countabilty of sets. We formulate it as the Countability Lemma. Suppose to each element of the set A there is assigned, by some definite rule, a unique natural number in such a manner that ...
... We draw attention to a simple principle, which can be used to prove many of the usual and important theorems on countabilty of sets. We formulate it as the Countability Lemma. Suppose to each element of the set A there is assigned, by some definite rule, a unique natural number in such a manner that ...
Honors Algebra 4, MATH 371 Winter 2010
... of the structure theorem for modules over a PID and part (8b). Solution: Since A satisfies its characteristic polynomial by Cayley-Hamilton, this characteristic polynomial annhihilates V as an F [X]-module. We deduce that V is torsion and hence isomorphic to F [X]/(a1 ) ⊕ · · · ⊕ F [X]/(ad ) for a1 ...
... of the structure theorem for modules over a PID and part (8b). Solution: Since A satisfies its characteristic polynomial by Cayley-Hamilton, this characteristic polynomial annhihilates V as an F [X]-module. We deduce that V is torsion and hence isomorphic to F [X]/(a1 ) ⊕ · · · ⊕ F [X]/(ad ) for a1 ...
Algebraic Number Theory Brian Osserman
... Kummer developed the theory of ideals of rings in part to prove the following remarkable theorem: Theorem 1.3.6. Suppose that p is an odd prime number such that p does not divide any of the numerators of B2 , B4 , . . . , Bp−3 (in particular, p = 3 is acceptable). Then the equation xp + y p = z p ha ...
... Kummer developed the theory of ideals of rings in part to prove the following remarkable theorem: Theorem 1.3.6. Suppose that p is an odd prime number such that p does not divide any of the numerators of B2 , B4 , . . . , Bp−3 (in particular, p = 3 is acceptable). Then the equation xp + y p = z p ha ...
PRIMITIVE ELEMENTS FOR p-DIVISIBLE GROUPS 1. Introduction
... We define primitive elements using non-nullity as follows. Assume that G = G[pr ], the pr -torsion subgroup of a p-divisible group G of height h. Let Gprim = G ×G[p] (G[p])× , where the map G → G[p] is given by multiplication by pr−1 . It follows that the subscheme Gprim ⊂ G is locally free over S o ...
... We define primitive elements using non-nullity as follows. Assume that G = G[pr ], the pr -torsion subgroup of a p-divisible group G of height h. Let Gprim = G ×G[p] (G[p])× , where the map G → G[p] is given by multiplication by pr−1 . It follows that the subscheme Gprim ⊂ G is locally free over S o ...
