Elements of Abstract Algebra (Dover Books on Mathematics)
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Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book.
The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult.
Georg Frobenius (1849–1917), and it is only fair to warn that the solution is somewhat lengthy.) Group Homomorphism and Isomorphism 60. A homomorphism of groups is a mapping from the set of elements of one group to the set of elements of another which preserves multiplication. In other words, a mapping φ : G → G is a group homomorphism if G and G′ are groups and if for all x, y ∈ G, φ(xy) = (φx)(φy). A group homomorphism φ : G → G is called an endomorphism (of the group G). The identity
Figure 13 The polynomial 8x3 − 6x − 1 is irreducible over Q: the substitution yields y3 − 3y2 − 3, which is clearly irreducible by the Eisenstein criterion (107); were 8x3 − 6x + 1 reducible over Q, the same substitution applied to its factors would yield a factorization of y3 − 3y2 − 3. Now it follows that 8x3 − 6x − 1 is a minimal polynomial for cos 20° over Q and that [Q(cos 20°): Q] = 3. By 120 we are forced to conclude that Q(cos 20°) is not a constructible number field and that cos
equation whose splitting field is associated with the symmetric group S5, which we know is not a solvable group. This elegant theory is the work of the tormented genius, Évariste Galois (1811–1832), whose brief life is the most tragic episode in the history of mathematics. Persecuted by stupid teachers, twice refused admission to the École Polytechnique, his manuscripts rejected, or even worse, lost by the learned societies, Galois in bitterness immersed himself in the radical politics of the
φn−1 are distinct automorphisms of E leaving Zp fixed and conclude that E is a Galois extension of Zp with cyclic Galois group 129. Theorem. E is a Galois extension of F if and only if the following conditions hold: an irreducible polynomial over F of degree m with at least one root in E has m distinct roots in E; E is a simple algebraic extension of F, that is, E = F(θ) for some element θ ∈ E which is algebraic over F. Proof. Necessity of condition (1). Suppose E is a Galois extension
152ε. Let R1 and R2 be rings. We define addition and multiplication on the cartesian product R1 x R2 by the rules (a1, a2) + (b1, b2) = (a1 + b1, a2 + b2) and (a1, a2)(b1, b2) = (a1b1, a2b2). Verify that R1 x R2 with these operations is a ring. (The ring obtained this way is called the direct product of R1 and R2 and denoted R1 x R2.) Is the direct product of integral domains an integral domain? 153. The Ring of Integers Modulo n. We already know (33) that Zn, the set of congruence