Fundamental Concepts of Abstract Algebra (Prindle, Weber, and Schmidt Series in Advanced Mathematics)
Gertrude Ehrlich
Language: English
Pages: 340
ISBN: 0534924557
Format: PDF / Kindle (mobi) / ePub
Beginning with sets, relations, and functions, the text proceeds to an examination of all types of groups, including cyclic groups, subgroups, permutation groups, normal subgroups, homomorphism, factor groups, and fundamental theorems. Additional topics include subfields, extensions, prime fields, separable extensions, fundamentals of Galois theory, and other subjects.
(the) n distinct n-th roots of 1 in . In the complex plane, by de Moivre’s Theorem, the points representing the h(h = 0,..., n − 1) are equally spaced around the unit circle, hence form the vertices of a regular n-gon. The primitive n-th roots of unity are the h with gcd (h, n) = 1. For example, the sixth roots of unity are as shown in Figure 15. The primitive sixth roots of unity are = e2i/6 and 5 = e10i/6 = e5i/3 = −1. ———————Figure 15——————— * * * In the following, we explore
and e′ ∈ A is a right identity for , then e′ = e. Hence A contains at most one two-sided identity for . Proof: e = e e′ = e′. Thus, if there is a (two-sided) identity for , in A, then this identity is unique, i.e., it is the only (two-sided) identity and, indeed, the only right or left identity, for in A. An identity element in a set A is characterized by its behavior toward every element of the set, with respect to a given binary operation. This is in sharp contrast to the behavior of
only if it has a normal series, all of whose factors are cyclic of prime order. Proof: Obviously, if G has a normal series all of whose factors are cyclic, then G is solvable. Conversely, suppose G is solvable. Let be a solvable normal series. Each Gi (i = 1,..., n) is finite; hence, between Gi and Gi−1, there are subgroups Hi1,..., such that where each of the Hij−1 is maximal normal in Hij (j = 1,..., si). By the Correspondence Theorem, it follows that, for each j = 1,..., Si, Hij/Hij−1
attention to a particularly interesting class of modules, known as vector spaces: they are modules of division rings and, in particular, of fields. Definition 3.11.1: Let F be a division ring. Then a unitary left F-module V is called a left vector space over V. Thus, explicitly, V is a left vector space over F if V, + is an abelian group, µ : F × V → V is an external composition (with µ(a, x) denoted by ax for each a ∈ F, x ∈ V) satisfying (1) a(x + y) = ax + ay for each a ∈ F, x, y ∈ V; (2)
have But then is the i-th column of M. Thus, for each is the i-th column of M. But then M = A. Example 1: * * * Let P3 = {f ∈ [x]|f = 0 or deg f < 3}. Define : P3 → P3 by (f) = f′ (the derivative of f). Let = ′ {1, x, x2}. Then Since (1) = 0 = 0.1 + 0x + 0x2 (x) = 1 = 1.1 + 0x + 0x2 (x2) = 2x = 0.1 + 2x + 0x2, We have Now let g = 2 + 3x + 5x2. Then and Thus, (g) = 3.1 + 10x, which we recognize as g′. * * * Example 2: * * * If we modify Example 1 by a change of