On sum-sets and product-sets of complex numbers
... Nathanson [9] proved the bound with ε = 1/31, Ford [8] improved it to ε = 1/15 , and the best bound is obtained by Elekes [6] who showed ε = 1/4 if A is a set of real numbers. Very recently Chang [3] proved ε = 1/54 to finite sets of complex numbers. For further results and related problems we refer ...
... Nathanson [9] proved the bound with ε = 1/31, Ford [8] improved it to ε = 1/15 , and the best bound is obtained by Elekes [6] who showed ε = 1/4 if A is a set of real numbers. Very recently Chang [3] proved ε = 1/54 to finite sets of complex numbers. For further results and related problems we refer ...
MATH 103B Homework 6 - Solutions Due May 17, 2013
... Recall that a PID is an integral domain with the additional property that every ideal in the ring is generated by some element in the ring. Proofs: I. We have proved that Z is an integral domain (cf. Chapter 13, example 1). Let A be an ideal of Z. If A “ t0u then A “ x0y so it is principal. Otherwi ...
... Recall that a PID is an integral domain with the additional property that every ideal in the ring is generated by some element in the ring. Proofs: I. We have proved that Z is an integral domain (cf. Chapter 13, example 1). Let A be an ideal of Z. If A “ t0u then A “ x0y so it is principal. Otherwi ...
![WHEN IS F[x,y] - American Mathematical Society](http://s1.studyres.com/store/data/017823178_1-6801a801f234da9ec7275765b0565209-300x300.png)
WHEN IS F[x,y] - American Mathematical Society
... Proposition 3. Let f be a linear polynomial in R = F[x, y] that is not associated to a central polynomial. Then ff is a C-atom. Proof. Let f = ax + by + c. If either a or b is 0 then / G F[y] or F[x], respectively, so ff is a C-atom by Proposition 2. Suppose now that both a and b are nonzero and // ...
... Proposition 3. Let f be a linear polynomial in R = F[x, y] that is not associated to a central polynomial. Then ff is a C-atom. Proof. Let f = ax + by + c. If either a or b is 0 then / G F[y] or F[x], respectively, so ff is a C-atom by Proposition 2. Suppose now that both a and b are nonzero and // ...
Fundamental Notions in Algebra – Exercise No. 10
... Definition: A ring R is called semi-primitive if for every element a 6= 0 of R there exists a simple R-module M such that a ∈ / Ann(M ). Definition: We say that aQring R is a subdirect product of rings Rα , if there exists an embedding ι : R → Rα such that the composition πα ◦ ι : R → Rα is surjecti ...
... Definition: A ring R is called semi-primitive if for every element a 6= 0 of R there exists a simple R-module M such that a ∈ / Ann(M ). Definition: We say that aQring R is a subdirect product of rings Rα , if there exists an embedding ι : R → Rα such that the composition πα ◦ ι : R → Rα is surjecti ...
non-abelian classfields over function fields in special cases
... 1.1. Primes and Conjugacy Classes Principle. One of our basic ideas is that a certain type of infinite discrete groups T plays a central role in arithmetic of non-abelian extensions of algebraic function fields of one variable over finite fields (abbrev. function fields). An origin of this idea was ...
... 1.1. Primes and Conjugacy Classes Principle. One of our basic ideas is that a certain type of infinite discrete groups T plays a central role in arithmetic of non-abelian extensions of algebraic function fields of one variable over finite fields (abbrev. function fields). An origin of this idea was ...
MATH882201 – Problem Set I (1) Let I be a directed set and {G i}i∈I
... (b) Give G the subspace topology. Show that G is compact and totally disconnected for this topology. (c) Prove that the natural projection maps φi : G → Gi are continuous, and that the (open) subgroups Ki := ker φi for a basis of open neighborhoods of the identity. (d) Show that a subgroup of G is o ...
... (b) Give G the subspace topology. Show that G is compact and totally disconnected for this topology. (c) Prove that the natural projection maps φi : G → Gi are continuous, and that the (open) subgroups Ki := ker φi for a basis of open neighborhoods of the identity. (d) Show that a subgroup of G is o ...
WHAT IS A POLYNOMIAL? 1. A Construction of the Complex
... • We describe the totality of polynomials having coefficients in R as an algebraic structure. The structure in question is a commutative R-algebra, meaning an associative, commutative ring A having scalar multiplication by R. (From now on in this writeup, algebras are understood to be commutative.) ...
... • We describe the totality of polynomials having coefficients in R as an algebraic structure. The structure in question is a commutative R-algebra, meaning an associative, commutative ring A having scalar multiplication by R. (From now on in this writeup, algebras are understood to be commutative.) ...
Class Notes: 9/1/09 MAE 501 Distributed by: James Lynch Structure
... Problem: Z is closed under multiplication but not Division. (We can multiply any two integers and get an integer but the same is not true for division) ...
... Problem: Z is closed under multiplication but not Division. (We can multiply any two integers and get an integer but the same is not true for division) ...
Problem Set 1 - University of Oxford
... 2. Let H denote the space of holomorphic (i.e. complex differentiable) functions f : C → C, and let C = {f : R → R : f is differentiable }. Which (if either) of H or C is an integral domain? Solution: Recall from complex analysis the Identity theorem: if a holomorphic function f : U → C defined on a ...
... 2. Let H denote the space of holomorphic (i.e. complex differentiable) functions f : C → C, and let C = {f : R → R : f is differentiable }. Which (if either) of H or C is an integral domain? Solution: Recall from complex analysis the Identity theorem: if a holomorphic function f : U → C defined on a ...
