By George R. Kempf (auth.)

The legislation of composition comprise addition and multiplication of numbers or func tions. those are the elemental operations of algebra. one could generalize those operations to teams the place there's only one legislations. the idea of this ebook was once began in 1800 through Gauss, while he solved the 2000 year-old Greek challenge approximately developing average n-gons by means of ruler and compass. the idea used to be extra built through Abel and Galois. After years of improvement the speculation used to be installed the current shape by means of E. Noether and E. Artin in 1930. at the moment it used to be referred to as sleek algebra and focused on the summary exposition of the speculation. these days there are too many examples to enter their information. i believe the scholar should still research the proofs of the theorems and never spend time searching for ideas to difficult routines. The routines are designed to explain the speculation. In algebra there are 4 simple buildings; teams, jewelry, fields and modules. We current the idea of those uncomplicated buildings. confidently this may provide an outstanding introduc tion to fashionable algebra. i've got assumed as historical past that the reader has discovered linear algebra over the genuine numbers yet this isn't necessary.

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**Example text**

So it = a I1O"EO(X - 0') = af where a E F. , the irreducible equation of e over F. 1. 0 The proof of this lemma contains interesting information not contained in the statement! Proof of Theorem We can now prove the only if part of the theorem. Let E = F(el, ... ,en). Let f be the product of the irreducible equations of the ei's over F. Then E is the splitting field of f by the calculation in the last proof. For the converse, assume that E is the splitting field of a separable polynomial in F[X].

11 Let N c M C P be submodules of P over a ring. Then P/M is isomorphic to (P/N)/(M/N) as modules. 2 BASES AND FREE MODULES Let S be a set and R be a ring. For each s in S we have an element ms of an R-module M. Then (ms)sEs spans M if any element m of M can be written as a finite sum L:s rsms where the rs are in R and all but a finite number of them are zero. ) Also (ms)sEs is a basis of M if it spans M and the expression m = L:s rsms is unique. 2 BASES AND FREE MODULES 27 We can write these definitions in terms of homomorphisms.

1 E ~ F(e) if and only if there are only finitely many fields between E and F. Proof. The "if' part is ... if F is finite then F( el, ... 3. Assume that F is infinite. Thus F(el, e2) contains only finitely many subfields ~ F. Thus for some Cl f. C2 in F we have F(el + cle2) = F(el + C2e2) = K. So e2 = [el + Cle2 - (el + C2e2)]' (Cl - C2)-1 E K, and el = (el + cle2) - Cle2 E K. Therefore E(el' e2) = K. By induction "if' is true. The "only if' is ... let E ~ G ~ F be a field. G(e). Let f E G[X] be the irreducible polynomial of e over G.