By George R. Kempf (auth.)

ISBN-10: 3322802787

ISBN-13: 9783322802781

ISBN-10: 3528065834

ISBN-13: 9783528065836

The legislation of composition comprise addition and multiplication of numbers or func tions. those are the fundamental operations of algebra. you'll be able to generalize those operations to teams the place there's only one legislations. the speculation of this ebook used to be began in 1800 by means of Gauss, while he solved the 2000 year-old Greek challenge approximately developing commonplace n-gons through ruler and compass. the idea was once additional built by way of Abel and Galois. After years of improvement the idea used to be installed the current shape by way of E. Noether and E. Artin in 1930. at the moment it was once referred to as glossy algebra and targeting the summary exposition of the speculation. these days there are too many examples to enter their information. i feel the scholar may still learn the proofs of the theorems and never spend time trying to find options to tough routines. The workouts are designed to elucidate the speculation. In algebra there are 4 uncomplicated buildings; teams, jewelry, fields and modules. We current the speculation of those easy buildings. optimistically it will provide a very good introduc tion to fashionable algebra. i've got assumed as history that the reader has realized linear algebra over the genuine numbers yet this isn't necessary.

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**Example text**

Assume that F is infinite. By induction on n, we may assume that n = 1. Let (31 = (3. Let f and 9 be the irreducible polynomial of a and (3 over F. Let G be a splitting field of fg over F containing E. Then f = 7l"i(X - ai) and 9 = 7l"k(X - (3k) where these (3k are distinct elements of G and (31 = (3 and a1 = a. If k =f 1, the equation ai + x(3k = a + x(3 has a unique solution. As F is infinite we may find r in F such that ai + r(3k =f a + r(3 for all k =f 1. Let 8 = a + r(3. I claim this works.

If we think we will see that these numbers determine nr,i, we have an old result. ) If R is a ring, let I and J be two ideals in R such that I + J = R. Then a) R/(I n J) is isomorphic to R/ I ffi R/ J and b) If A is commutative In J = I· J where I· J = {L: Cs . js where Cs E I and js E J}. Proof. We have a homomorphism cp : R ---+ R/ I ffi R/ J sending r to (r + I, r + J). Clearly Ker cp = In J. We need to see that cp is surjective. We have 1 = Yl + Y2 for some Yl E I, Y2 E J. Let (Xl + I, X2 + J) be an element of R/ I ffi R/ J.

The elements in the lemma are contained 0 in F(el)(e2). 2. Let FeE be a field extension. The point of the lemma is that the set {e E Ele is algebraic over F} is a subfield of E which contains F. The classical case is for the extension Q C C. Then an element of the set is called an algebraic number. Many people have studied algebraic numbers beginning with the algebra of this book. 1 In the situation of the lemma, show that any polynomial in el and e2 with coefficients in F has degree over F ~ (deg el over F) (deg e2 over F).

### Algebraic Structures by George R. Kempf (auth.)

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