Section 2.1
... Factorization in Polynomial Rings 2.1 Polynomial Rings If R is any ring, we can define the ring R[x] of polynomials with coefficients in R. If F is a field, then the polynomial ring F[x] is a particular interest. Polynomial rings over fields have some resemblance to the ring Z of integers in terms o ...
... Factorization in Polynomial Rings 2.1 Polynomial Rings If R is any ring, we can define the ring R[x] of polynomials with coefficients in R. If F is a field, then the polynomial ring F[x] is a particular interest. Polynomial rings over fields have some resemblance to the ring Z of integers in terms o ...
LECTURE 1: REPRESENTATIONS OF SYMMETRIC GROUPS, I 1. Introduction S
... “traditional” approach based on Young symmetrizers, the reader is welcome to consult [E] or [F]. 2. Inductive approach A key observation is that symmetric groups for different n are embedded into one another: {1} = S1 ⊂ S2 ⊂ S3 . . . ⊂ Sn−1 ⊂ Sn ⊂ . . ., where we view Sn−1 as the subgroup of Sn fixin ...
... “traditional” approach based on Young symmetrizers, the reader is welcome to consult [E] or [F]. 2. Inductive approach A key observation is that symmetric groups for different n are embedded into one another: {1} = S1 ⊂ S2 ⊂ S3 . . . ⊂ Sn−1 ⊂ Sn ⊂ . . ., where we view Sn−1 as the subgroup of Sn fixin ...
selected solutions to Homework 11
... / S. Thus, S is not closed under addition and is not a subring. (b) This statement is true. Proof. Let R be a ring with unity, 1. Let a, b ∈ S. This means that there exist c, d ∈ R such that ac = 1 and bd = 1. Then (ab)(dc) = a(bd)c = a(1)(c) = (ac) = 1 Thus, (ab) ∈ S and S is closed under multiplic ...
... / S. Thus, S is not closed under addition and is not a subring. (b) This statement is true. Proof. Let R be a ring with unity, 1. Let a, b ∈ S. This means that there exist c, d ∈ R such that ac = 1 and bd = 1. Then (ab)(dc) = a(bd)c = a(1)(c) = (ac) = 1 Thus, (ab) ∈ S and S is closed under multiplic ...
Math 581 Problem Set 1 Solutions
... set of k + 1 elements, say B = {b1 , . . . , bk , bk+1 }. We split the injective functions into k + 1 sets Ai where the functions in Ai are the injective functions that send b1 to bi . The functions in the set A1 send b1 to b1 , so are determined by what the function does on the set of k elements {b ...
... set of k + 1 elements, say B = {b1 , . . . , bk , bk+1 }. We split the injective functions into k + 1 sets Ai where the functions in Ai are the injective functions that send b1 to bi . The functions in the set A1 send b1 to b1 , so are determined by what the function does on the set of k elements {b ...
Group action
... 1. (a) Assume that prime number p divides x16 + 1. Then p is of form 32k + 1. (b) Without using the general Dirichlet theorem or L-functions, prove that for any k there is infinite set of primes of form kn + 1. Solution. (a) Modulu p, the equation x16 = –1 has a solution. So, there is a remainder x ...
... 1. (a) Assume that prime number p divides x16 + 1. Then p is of form 32k + 1. (b) Without using the general Dirichlet theorem or L-functions, prove that for any k there is infinite set of primes of form kn + 1. Solution. (a) Modulu p, the equation x16 = –1 has a solution. So, there is a remainder x ...
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements together with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation; the addition of any two integers forms another integer. The abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way, while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Point groups are used to help understand symmetry phenomena in molecular chemistry; and Poincaré groups can express the physical symmetry underlying special relativity.The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups announced in 1983. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become a particularly active area in group theory.