By Y. Bahturin

ISBN-10: 9048143179

ISBN-13: 9789048143177

ISBN-10: 9401708398

ISBN-13: 9789401708395

This publication has constructed from a chain of lectures that have been given through the writer in mechanics-mathematics division of the Moscow kingdom college. In 1981 the direction "Additional chapters in algebra" changed the direction "Gen eral algebra" which used to be based through A. G. Kurosh (1908-1971), professor and head of the dep. of upper algebra for a interval of numerous a long time. the cloth of this direction shaped the foundation of A. G. Kurosh's recognized e-book "Lectures on basic algebra" (Moscow,1962; 2-nd variation: Moscow, Nauka, 1973) and the ebook "General algebra. Lectures of 1969-1970. " (Moscow, Nauka, 1974). one other booklet in line with the path, "Elements of basic al gebra" (M. : Nauka, 1983) used to be released via L. A. Skorniakov, professor, now deceased, within the related division. it's going to be famous undefined. G. Kurosh used to be not just the lecturer for the path "General algebra" yet he was once additionally the well-known chief of the medical institution of an analogous identify. it's tricky to figure out the bounds of this college; in spite of the fact that, the "Lectures . . . " of 1962 males tioned above comprise a few fabric which exceed those limits. ultimately this impact intensified: the lectures of the path got by means of many famous scientists, and a few of them see themselves as "general algebraists". every one lecturer introduced major originality not just in presentation of the cloth yet within the substance of the direction. consequently now not all fabric that is now authorised as priceless for algebraic scholars suits in the scope of common algebra.

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E. I = Rq(t). Since p(t) E I we get p{t) = s(t)q(t), for some s(t). One of q(t), s(t) is invertible and the other one is associated to p{t). If q(t) is invertible then I = Rq(t) = R. If q(t) = fP{t) is associated to p{t) we have I = RfP{t) = Rp(t) = M. Thus M is a maximal ideal. Now, put F = RIM; according to the proposition above F is a field. In addition it should be mentioned that the field F can be identified with the subfield in F consisting of cosets of polynomials of zero degree ~ E F +-+ X= ~ + M E F.

Let Cl and C2 be two algebraic closures of the field F. It is useful to note the following fact which plays an important role in the proof of the uniqueness of algebraic closure. Lemma. Let K be an algebraic extension of the field F, Q an algebraically closed field. Then any imbedding ¢ : F --T Q extends to an imbedding ¢; : K --T Q for which ¢;IF = ¢. Proof. We shall use the Zorn lemma. e. "pIF = ¢. We put that (E,,,p) S; (E',,,p') if E c E' and "p'IE = "p. The set II is non--empty as the pair (F, ¢) is an element.

According to the Schur lemma, 7r1 is an isomorphism and u is equivalent to PI. Thus any irreducible representation is equivalent to one of PI, ... ,P3. 0 § 3. 10. Character of a representation. Let V be the space of a finitedimensional representation P of the group G over a field F. We define the character of the representation as the function X : G ----+ F such that X(g) = tr peg). * We should recall that the trace tr A of the linear operator A : V ----+ V is a sum of the diagonal elements of its matrix in some (in fact, in any) basis of the space V.

### Basic Structures of Modern Algebra by Y. Bahturin

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