Algebraic Structures by George R. Kempf (auth.)

By George R. Kempf (auth.)

The legislation of composition contain addition and multiplication of numbers or func­ tions. those are the elemental operations of algebra. you possibly can generalize those operations to teams the place there's only one legislation. the idea of this booklet was once all started in 1800 by means of Gauss, whilst he solved the 2000 year-old Greek challenge approximately developing normal n-gons by way of ruler and compass. the idea was once additional built through Abel and Galois. After years of improvement the idea was once installed the current shape via E. Noether and E. Artin in 1930. at the moment it used to be referred to as smooth algebra and focused on the summary exposition of the idea. these days there are too many examples to enter their info. i believe the scholar may still research the proofs of the theorems and never spend time trying to find options to tough routines. The workouts are designed to explain the speculation. In algebra there are 4 uncomplicated buildings; teams, earrings, fields and modules. We current the idea of those uncomplicated constructions. confidently it will supply an outstanding introduc­ tion to trendy algebra. i've got assumed as heritage that the reader has discovered linear algebra over the genuine numbers yet this isn't necessary.

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Let ell ... , en be elements of E which have separable irreducible polynomials in F[X). If FeE is a field extension, then E is separable over F if each element of E is separable over F. 1 There is an element e of E which is separable over F such that F(e) = F(el"'" en). Proof. By induction we need only prove the case n = 2. We can now finish Galois theory. 3. 3 Let H be a subgroup of Aut(E/F). Let G = {e E Elh(e) = e for all h E H}. Then F C GeE and E/G is Galois. Then E is the splitting field of a separable polynomial f in G[X].

If e E E, then let fe be its irreducible polynomial of e over F. Let R be the splitting field of feI . fe2 over F containing F(el, e2). Then F(el, e2) is contained in Rand R is Galois over F. 2 fe is separable where e is any element of R, in particular if e E F(el, e2). 1 If w = ~(-1 + V3i) (a cubic root of unity) a) find the degree of Q( w, ij2) over Q, b) show that it is a Galois extension of Q, and c) find a number a such that Q(w, ij2) = Q(a). Let ell ... , en be elements of E which have separable irreducible polynomials in F[X).

Then Ker K is a prime ideal. There are two cases. CASE 1. (d = 0). Then Z C F and hence Q C F. In this we say F has characteristic zero. CASE 2. (d = prime p). Then Z/pZ C F and Z/pZ is a field with p elements. In this case we say that F has characteristic p. Sometimes Q or Z/pZ are called the prime field. Let FeE be a subfield of a field E. Let e be an element of E. We have a homomorphism '¢e : F[X] - t E sending X to e and the identity on constants. Then ker '¢e = gF[X] where 9 is a polynomial in F[X].

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