By George R. Kempf (auth.)

ISBN-10: 3322802787

ISBN-13: 9783322802781

ISBN-10: 3528065834

ISBN-13: 9783528065836

The legislation of composition contain addition and multiplication of numbers or func tions. those are the elemental operations of algebra. you can still generalize those operations to teams the place there's only one legislations. the idea of this booklet used to be began in 1800 by means of Gauss, while he solved the 2000 year-old Greek challenge approximately developing usual n-gons through ruler and compass. the speculation used to be extra constructed via Abel and Galois. After years of improvement the idea used to be installed the current shape through E. Noether and E. Artin in 1930. at the moment it was once known as sleek algebra and focused on the summary exposition of the speculation. these days there are too many examples to enter their information. i feel the coed may still research the proofs of the theorems and never spend time trying to find suggestions to difficult routines. The workouts are designed to explain the idea. In algebra there are 4 uncomplicated constructions; teams, earrings, fields and modules. We current the speculation of those uncomplicated constructions. optimistically this can supply a great introduc tion to fashionable algebra. i've got assumed as heritage that the reader has realized linear algebra over the genuine numbers yet this isn't necessary.

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**Example text**

I/J)~ = 2:l~p~j

B) Same with lI aE !. If M is an R-module, HomR(M, R) is dual to M and its denoted by MD. 4 If M is isomorphic to R n where R is commutative, then its dual MD is isomorphic to Rn as an R-module. 1 SYLOW'S THEOREMS. Let G be a finite group. Let p be a prime. Assume that #G = pb· m where (p, m) = 1. 1 If pal#G, then there is a subgroup H C G with pa = #H. Proof. Let X be the set of subsets L of G with pa elements. Then #X = (pb . m)(pb . m - 1) ... (pbm - pa + 1) (pa)(pa - 1) ... :... because the power of p dividing pb.

Let F' = {e E Elrp(e) = e for all rp in Aut(E/F)}. Clearly F' is a subfield of E containing F. Define the field extension FeE is Galois if E is finite over F and F' = F. Thus Galois extensions are very symmetric. Here Aut(E / F) is called the Galois group of the Galois field extension FeE. 2 R c C is a Galois extension. 1 FeE is Galois if and only if E is a splitting field of a separable polynomial in F[X]. Let E / F be Galois. 1 Aut(E/F) is finite. Proof. Let E = F( e1, ... , ed). Let rp E A ut( E / F).

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Categories: Algebra