tailieunhanh - A First Course In Abstract Algebra-Jb Fraleigh, 7Ed 2

(BQ) Part 2 book "A first course in abstract algebra" has contents: Extention fields, advanced group theory, groups in topology, factorization, automorphisms and galois theory. | PART VI Extension Fields Section 29 Introduction to Extension Fields Section 30 Vector Spaces Section 31 Algebraic Extensions Section 32 Geometric Constructions Section 33 Finite Fields SECTION 29 Introduction to Extension Fields Our Basic Goal Achieved We are now in a position to achieve our basic goal which loosely stated is to show that every nonconstant polynomial has a zero. This will be stated more precisely and proved in Theorem . We first introduce some new terminology for some old ideas. Definition A field E is an extension field of a field F if F E. Figure Thus R is an extension field of Q and c is an extension field of both R and Q. As in the study of groups it will often be convenient to use subfield diagrams to picture extension fields the larger field being on top. We illustrate this in Fig. . A configuration where there is just one single column of fields as at the left-hand side of Fig. is often referred to without any precise definition as a tower of fields. i Section 32 is not requừed for the remainder of the text. 265 266 Part VI Extension Fields Now for our basic goal This great and important result follows quickly and elegantly from the techniques we now have at our disposal. Theorem Kronecker s Theorem Basic Goal Let F be a field and let x be a nonconstant polynomial in F x . Then there exists an extension field E of F and an a 6 E such that a 0. Proof By Theorem f x has a factorization in F x into polynomials that are irreducible over F. Let p x be an irreducible polynomial in such a factorization. It is clearly sufficient to find an extension field E of F containing an element a such that p a 0. By Theorem p x is a maximal ideal in F x so F x p x is a field. We claim that F can be identified with a subfield of F x Xr in a natural way by use of the map Ỷ F F x p x given by Ỷ a a p x fora e F. This map is one to one for if ỉịr a i r è that is if a p x b p x for some a b e F then a - b e p x so a b must .