By Falko Lorenz

From Math experiences: "This is a captivating textbook, introducing the reader to the classical elements of algebra. The exposition is admirably transparent and lucidly written with merely minimum necessities from linear algebra. the recent strategies are, a minimum of within the first a part of the e-book, outlined within the framework of the advance of rigorously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on buildings with ruler and compass of their algebraic surroundings within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The booklet ends with an appendix containing workouts and notes at the past elements of the publication. notwithstanding, short historic reviews and proposals for extra examining also are scattered in the course of the text."

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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 . F13. I is a maximal ideal of R if and only if the quotient ring R=I is simple. Proof. Ideals of R containing I are in one-to-one correspondence, via the quotient map W R !

EM /M 2I , to conclude that there exists a ﬁeld extension E=K and K-homomorphisms M W EM ! E for each M 2 I . We claim that for every ﬁnite ﬁeld extension L=K there exists a K-homomorphism from L into E. ˛1 ; : : : ; ˛m / with each ˛i algebraic over K, consider the homomorphism of K-algebras ' W KŒX1 ; : : : ; Xm ! Xi / D ˛i . Let M be its kernel. Then ' yields an isomorphism KŒX1 ; : : : ; Km =M ! L. Hence M is a maximal ideal of KŒX1 ; : : : ; Xm , and the claim is proved. Now let C be the algebraic closure of K in E (see Chapter 2, F8).

A/. ˜ A study of this proof leads to the following generalization: Deﬁnition 4. An integral domain R is called a Euclidean domain if there exists a map W R ! 0/ D 0 and that, for every a; b 2 R with a ¤ 0, there exist q; r 2 R with b D qa C r Such a map Examples. a/: is called a Euclidean valuation on R. a/ D jaj. deg g/ C 1 for g ¤ 0. deﬁned by F6. Every Euclidean domain R is a principal ideal domain. This is proved exactly like the case R D ޚof F5. 3. We now generalize the familiar notion of prime numbers in ޚand irreducible polynomials in KŒX .