By Falko Lorenz

From Math studies: "This is a captivating 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 options are, not less than within the first a part of the booklet, outlined within the framework of the improvement of rigorously chosen difficulties. therefore, 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 ebook ends with an appendix containing workouts and notes at the past elements of the e-book. although, short ancient reviews and proposals for extra analyzing also are scattered throughout the text."

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**Extra resources for Algebra: Volume I: Fields and Galois Theory**

**Example text**

Deﬁnition 2. X / 2 C ŒX has a root in C . Remark. The ﬁeld ރof complex number is algebraic closed; this is proved through analysis, function theory or algebra (see Volume II for the latter). F2. The following statements about a ﬁeld C are equivalent: (i) C is algebraically closed. (ii) Every irreducible polynomial in C ŒX is linear (that is, of degree 1). (iii) Every nonconstant polynomial in C ŒX is completely decomposable into linear factors. (iv) If E=C is an algebraic ﬁeld extension, E D C .

It turns out to be useful to extend our terminology a little in a formal sense. Toward this goal we ﬁrst ﬁx a directory of primes ᏼ of R specifying a representative for each class of associated primes; that is, ᏼ is a set of nonzero prime elements of R such that every nonzero prime of R is associated with one and only one element of ᏼ. ) In many cases there is a canonical choice for ᏼ — for example, in R D ޚthe set of natural prime numbers stands out, and in the polynomial ring KŒX over a ﬁeld K we can take for ᏼ the set of all normalized prime polynomials.

We regard EM as an extension of K. 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.