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 purely minimum must haves from linear algebra. the recent ideas are, at the least within the first a part of the e-book, outlined within the framework of the improvement of rigorously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on buildings 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 publication ends with an appendix containing workouts and notes at the earlier components of the publication. although, short historic reviews and proposals for additional analyzing also are scattered during the text."

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Example text

Any two gcd’s of a1 ; : : : ; an are associated to one another. Likewise for any two lcm’s of a1 ; : : : ; an . This follows immediately from the definitions. But how about the existence of a gcd or lcm for given elements of R ? 2. a/: This translation of the divisibility relation into a simple inclusion relation is very fruitful. b/. b/ is a principal ideal. F3. Let I1 and I2 be ideals of R. Then I1 \ I2 and I1 C I2 WD f˛1 C ˛2 j ˛i 2 Ii g are also ideals of R; and in fact I1 \ I2 is the largest ideal of R contained in I1 and I2 , and I1 C I2 is the smallest ideal of R containing I1 and I2 .

Proof. Let Ä W R ! E be an injective ring homomorphism of R into a field E. We define W F ! Ãa=Ãb/ D Äa=Äb: It is easy to check that is well defined. It is also clear that is a ring homomorphism, and that in fact it’s the only one for which diagram (23) commutes. ˜ Proof of F7. We first show the uniqueness statement. By assumption there exist homomorphisms W F ! F 0 and 0 W F 0 ! F with ı Ã D Ã0 and 0 ı Ã0 D Ã. It follows that 0 ı ı Ã D 0 ı Ã0 D Ã, and thus, because of the uniqueness requirement, 0 ı D idF ; analogously we have ı 0 D idF 0 .

D 0, so ˇ 3 D 2. This contradicts the fact that 2 does not lie in ‫ ޑ‬. ) Taking into account equation (7) of Chapter 2, we immediately get the corollary p 3 ‫ ޑ‬. 2/ W ‫ ޑ‬D 3: As a first fruit of our algebraic study of the constructibility problem, we obtain from this and from Chapter 1, F9: p 3 2 is not constructible from f0; 1g with ruler and compass. ConseTheorem 3. quently, the Delian problem of the doubling of the cube is also not soluble. Here is an important field-theoretical application of the results from this chapter: Theorem 4 (Kronecker).

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