By Falko Lorenz
From Math reports: "This is an enthralling textbook, introducing the reader to the classical components of algebra. The exposition is admirably transparent and lucidly written with basically minimum necessities from linear algebra. the hot suggestions are, not less than within the first a part of the booklet, outlined within the framework of the improvement of rigorously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on structures with ruler and compass of their algebraic atmosphere within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The e-book ends with an appendix containing workouts and notes at the past elements of the e-book. despite the fact that, short ancient reviews and recommendations for additional analyzing also are scattered during the text."
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Additional info for Algebra: Volume I: Fields and Galois Theory
1 ; : : : ; ˛n / with ﬁnitely many elements ˛i ; otherwise there would be an inﬁnite chain of intermediate ﬁelds obtained by adjoining ever more elements. ˛; ˇ/. Since ᐆ is ﬁnite but K is inﬁnite, there exist distinct 1 ; 2 2 K such that K. 1 ˛ C ˇ/ D K. 2 ˛ C ˇ/ DW L: Then . 1 ˛ C ˇ/ . 2 ˛ C ˇ/ D . 1 2 /˛ lies in L, and therefore so does ˛, and likewise ˇ. It follows that E D L D K. 1 ˛ C ˇ/, so that E=K is simple (with D 1 ˛ C ˇ as a primitive element). ˛; ˛n /. For K a ﬁnite ﬁeld the assertion follows from the fundamental theorem of the theory of ﬁnite ﬁelds, which we will study later (Theorem 2 in Chapter 9).
Let K be a ﬁeld. (I) There exists an extension C of K with the following properties: (i) C is algebraically closed. (ii) C=K is algebraic. Such a ﬁeld is called an algebraic closure of K. (II) If C1 and C2 are algebraic closures of K, the extensions C1 =K and C2 =K are isomorphic. Proof of part (I). Let KŒXn ; n 2 ގbe the polynomial ring in countably many indeterminates X1 ; X2 ; : : : over K. Consider the set I of all subsets M Â KŒXn ; n 2 ގ such that M is a maximal ideal of KŒX1 ; : : : ; Xm for some m 2 ގ: For each such M , let EM WD KŒX1 ; : : : ; Xm =M be the corresponding quotient ﬁeld.
Consider the kernel I of the substitution homomorphism ' W RŒX ! X / D 0. X /. Thus (1) RŒX =X D RŒX =I ' R is an integral domain, that is, X is prime in RŒX , by Deﬁnition 9 of Chapter 4. But then the next statement shows that RŒX =X is actually a ﬁeld, and therefore by (1) R is also a ﬁeld. ˜ F2. If A is a principal ideal domain and ﬁeld. is irreducible in A, the quotient A= is a Proof. According to F14 in Chapter 4, we have to show that . / is a maximal ideal of A. Let I be a proper ideal of A containing .
Algebra: Volume I: Fields and Galois Theory by Falko Lorenz