Basic Algorithmic Number Theory
... Let [0, 1] = {x ∈ R : 0 ≤ x ≤ 1}. A function p : N → [0, 1] is overwhelming if 1 − p(κ) is negligible. Note that noticeable is not the logical negation of negligible. There are functions that are neither negligible nor noticeable. Example 2.1.11. The function ǫ(κ) = 1/2κ is negligible. Exercise 2.1. ...
... Let [0, 1] = {x ∈ R : 0 ≤ x ≤ 1}. A function p : N → [0, 1] is overwhelming if 1 − p(κ) is negligible. Note that noticeable is not the logical negation of negligible. There are functions that are neither negligible nor noticeable. Example 2.1.11. The function ǫ(κ) = 1/2κ is negligible. Exercise 2.1. ...
Integral domains in which nonzero locally principal ideals are
... that for some nonzero x ∈ I, Dx has a primary decomposition. Then I is invertible. Thus an integral domain in which every proper principal ideal has a primary decomposition is an LPI domain. Proof. (a) (1) ⇒ (2) ⇒ (3) ⇒ (4) Clear. (4) ⇒ (1) Let M be a maximal ideal of D. Now ((Dx : I)I)M = (DM x : I ...
... that for some nonzero x ∈ I, Dx has a primary decomposition. Then I is invertible. Thus an integral domain in which every proper principal ideal has a primary decomposition is an LPI domain. Proof. (a) (1) ⇒ (2) ⇒ (3) ⇒ (4) Clear. (4) ⇒ (1) Let M be a maximal ideal of D. Now ((Dx : I)I)M = (DM x : I ...
Discrete Mathematics - Lecture 8: Proof Technique (Case Study)
... Sometimes the problem can be split into smaller problems that can be easier to tackle individually. Sometimes viewing the problem is a different way can also help in tackling the problem easily. Whether to split a problem or how to split a problem or how to look at a problem is an ART that has to be ...
... Sometimes the problem can be split into smaller problems that can be easier to tackle individually. Sometimes viewing the problem is a different way can also help in tackling the problem easily. Whether to split a problem or how to split a problem or how to look at a problem is an ART that has to be ...
ENDOMORPHISMS OF ELLIPTIC CURVES 0.1. Endomorphisms
... of its Galois conjugates E σ . Most non-CM elliptic curves do not have this property. The ones that do are called (confusingly enough) Q-curves, and they have an important role to play in 21st century number theory. Exercise 1.5*: It follows from the facts recalled in the last section (and the primi ...
... of its Galois conjugates E σ . Most non-CM elliptic curves do not have this property. The ones that do are called (confusingly enough) Q-curves, and they have an important role to play in 21st century number theory. Exercise 1.5*: It follows from the facts recalled in the last section (and the primi ...
Elliptic Curves with Complex Multiplication and the Conjecture of
... The purpose of these notes is to present a reasonably self-contained exposition of recent results concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. The goal is the following theorem. Theorem. Suppose E is an elliptic curve defined over an imaginary ...
... The purpose of these notes is to present a reasonably self-contained exposition of recent results concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication. The goal is the following theorem. Theorem. Suppose E is an elliptic curve defined over an imaginary ...
Commutative ideal theory without finiteness
... a prime integer and n is an integer. Thus for R = Z every nonzero proper Qirreducible R-submodule of Q is a fractional ideal of a valuation overring of R. Moreover, every nonzero fractional R-ideal has a unique representation as an irredundant intersection of infinitely many completely Q-irreducible ...
... a prime integer and n is an integer. Thus for R = Z every nonzero proper Qirreducible R-submodule of Q is a fractional ideal of a valuation overring of R. Moreover, every nonzero fractional R-ideal has a unique representation as an irredundant intersection of infinitely many completely Q-irreducible ...