Algebra: Fields and Galois Theory by Falko Lorenz

By Falko Lorenz

From Math studies: "This is an enthralling textbook, introducing the reader to the classical components of algebra. The exposition is admirably transparent and lucidly written with in basic terms minimum must haves from linear algebra. the hot ideas are, no less than within the first a part of the e-book, outlined within the framework of the improvement of conscientiously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on structures with ruler and compass of their algebraic environment within the first bankruptcy introduces the reader spontaneously to such basic algebraic notions as box extension, the measure of an extension, etc... The booklet ends with an appendix containing routines and notes at the past components of the ebook. notwithstanding, short ancient reviews and recommendations for additional analyzing also are scattered in the course of the text."

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Part (iv) now is an automatic consequence of (iii). ˜ 5. The foregoing sections have dealt with little more than the general foundations of elementary arithmetic. 2. Definition 8. Let R be a (not necessarily commutative) ring with unity 1 ¤ 0. We call R simple if every homomorphism R ! R0 into an arbitrary ring R0 is either injective or the zero map. 2) a ring R (with 1 ¤ 0) is simple if and only if f0g and R are the only ideals of R. An ideal I ¤ R of R is a maximal ideal of R if there is no ideal of R distinct from I and R and containing I .

Remark. A field homomorphism W E1 ! 1/ D 1 by definition; therefore it is always injective and so gives rise to an isomorphism of E1 with a subfield of E2 . c/ D c for all c 2 K: From Definition 1 there is a steep but well-traveled path to Galois theory (opened largely by Dedekind and E. Artin; see the latter’s Galois Theory). Here we will take the more leisurely and scenic route. The following result is simple but far-reaching: F1. Let E=K and E 0=K 0 be field extensions and W K ! K 0 a field homomorphism.

W L (see Chapter 2, F3). ˛/ D E, so (32) says simply that E W F D E W L. By the degree formula this means F W K D L W K, which (since F  L) demands that F D L. ˜ 32 3 Simple Extensions Theorem 5. Suppose E=K is an algebraic extension. Then E=K is simple if and only if it possesses only finitely many intermediate fields. Proof. Denote by ᐆ be the set of all intermediate fields of E=K. ˛/. To prove the finiteness of ᐆ, consider the set ˇ ˚ « Ᏸ D g 2 EŒX  ˇ g is normalized and divides f in EŒX  : Now, it is well known that EŒX  enjoys unique factorization into prime factors (see for example LA II, p.

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