Introduction to Algebra by Hartmut Laue

By Hartmut Laue

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We then have {1X , x, x2 , . . , xo(x)−1 } = Zϕx ≤ X. 2. Let X be a finite group. Then o(x) = min{k|k ∈ N, xk = 1X } x ∈ X. |X| for all If n = 0, x is called of infinite order. For all T ⊆ X we put T := H≤X T ⊆H H. As an intersection of a non-empty set of subgroups of X, T is a subgroup of X and obviously the smallest subgroup containing T , called the subgroup generated by T . If T = {x1 , . . , xk }, it is also denoted by x1 , . . , xk . 3 Definition. A group X is called cyclic if there exists an element x ∈ X such that x = X.

Zkj ∈ Z, a1 , . . , akj ∈ A such that bj = (j) i∈kj (j) ai • zi . It follows that (j) v= j∈n i∈kj (j) ai • zi ◦ yj = (j) j∈n i∈kj aji ◦ (zi ◦ yj ) ∈ Z ◦ Y A . (2) The mapping Z × Y → Z ◦ Y , (z, y) → z ◦ y is surjective by definition of Z ◦ Y . If z, z ′ ∈ Z, y, y ′ ∈ Y such that z ◦ y = z ′ ◦ y ′, then z ◦ y − z ′ ◦ y ′ = 0V , hence y = y ′ , z = z ′ as Y is B-independent and z, z ′ = 0B . Now let U be a finite subset of Z ◦ Y . For each y ∈ Y set Zy := {z|z ∈ Z, z ◦ y ∈ U}, and for every pair (z, y) ∈ Z × Y such that z ◦ y ∈ U let az,y ∈ A.

A, b) ∼ (a′ , b′ ). 2 it follows inductively that, for any integral domain R, the polynomial ring in n variables R[t1 , . . , tn ] is an integral domain. If Q is a quotient field of R, we write Q(t1 , . . , tn ) for the quotient field of R[t1 , . . , tn ]. Its elements are commonly written as fractions fg where f, g ∈ R[t1 , . . , tn ], g = 0R . In particular, if K is a field, K(t1 , . . , tn ) stands for the quotient field of K[t1 , . . 49 We know that every π ∈ Sn determines The image of an element x ∈ Q is given only in dependence of the elements a ∈ R, b ∈ R˙ chosen to represent x in the form a · b−1 .

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